In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, nonstandard calculus is the modern application of
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
s, in the sense of
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
, to infinitesimal
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. It provides a rigorous justification for some arguments in calculus that were previously considered merely
heuristic
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
.
Non-rigorous calculations with infinitesimals were widely used before
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
sought to replace them with the
(ε, δ)-definition of limit
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
starting in the 1870s. (See
history of calculus
Calculus, originally called infinitesimal calculus, is a mathematical discipline focused on limits, continuity, derivatives, integrals, and infinite series. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, a ...
.) For almost one hundred years thereafter, mathematicians such as
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
viewed infinitesimals as being naive and vague or meaningless.
Contrary to such views,
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by
Edwin Hewitt
Edwin Hewitt (January 20, 1920, Everett, Washington – June 21, 1999) was an American mathematician known for his work in abstract harmonic analysis and for his discovery, in collaboration with Leonard Jimmie Savage, of the Hewitt–Savage z ...
and
Jerzy Łoś
Jerzy Łoś (born 22 March 1920 in Lwów, Poland (now Lviv, Ukraine) – 1 June 1998 in Warsaw) () was a Polish mathematician, logician, economist, and philosopher. He is especially known for his work in model theory, in particular for " Łoś's th ...
. According to
Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century."
History
The history of nonstandard calculus began with the use of infinitely small quantities, called
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
s in
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. The use of infinitesimals can be found in the foundations of calculus independently developed by
Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...
and
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
starting in the 1660s.
John Wallis
John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...
refined earlier techniques of
indivisibles
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows:
* 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...
of
Cavalieri and others by exploiting an
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
quantity he denoted
in area calculations, preparing the ground for integral
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. They drew on the work of such mathematicians as
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
,
Isaac Barrow
Isaac Barrow (October 1630 – 4 May 1677) was an English Christian theologian and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for proof of the fundamental theorem ...
and
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...
.
In early calculus the use of
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
quantities was criticized by a number of authors, most notably
Michel Rolle
Michel Rolle (21 April 1652 – 8 November 1719) was a French mathematician. He is best known for Rolle's theorem (1691). He is also the co-inventor in Europe of Gaussian elimination (1690).
Life
Rolle was born in Ambert, Basse-Auvergne. Rol ...
and
Bishop Berkeley
George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immateri ...
in his book ''
The Analyst''.
Several mathematicians, including
Maclaurin
Maclaurin or MacLaurin is a surname. Notable people with the surname include:
* Colin Maclaurin (1698–1746), Scottish mathematician
* Normand MacLaurin (1835–1914), Australian politician and university administrator
* Henry Normand MacLaurin ( ...
and
d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the ''Encyclopédie ...
, advocated the use of limits.
Augustin Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
developed a versatile spectrum of foundational approaches, including a definition of
continuity in terms of infinitesimals and a (somewhat imprecise) prototype of an
ε, δ argument in working with differentiation.
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
formalized the concept of
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
in the context of a (real) number system without infinitesimals. Following the work of Weierstrass, it eventually became common to base calculus on ε, δ arguments instead of infinitesimals.
This approach formalized by Weierstrass came to be known as the ''standard'' calculus. After many years of the infinitesimal approach to calculus having fallen into disuse other than as an introductory pedagogical tool, use of infinitesimal quantities was finally given a rigorous foundation by
Abraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincorpo ...
in the 1960s. Robinson's approach is called
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
to distinguish it from the standard use of limits. This approach used technical machinery from
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
to create a theory of
hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
s that interpret infinitesimals in a manner that allows a Leibniz-like development of the usual rules of calculus. An alternative approach, developed by
Edward Nelson
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical ...
, finds infinitesimals on the ordinary real line itself, and involves a modification of the foundational setting by extending
ZFC through the introduction of a new unary predicate "standard".
Motivation
To calculate the derivative
of the function
at ''x'', both approaches agree on the algebraic manipulations:
:
This becomes a computation of the derivatives using the
hyperreals
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
if
is interpreted as an infinitesimal and the symbol "
" is the relation "is infinitely close to".
In order to make ''f a real-valued function, the final term
is dispensed with. In the standard approach using only real numbers, that is done by taking the limit as
tends to zero. In the
hyperreal
Hyperreal may refer to:
* Hyperreal numbers, an extension of the real numbers in mathematics that are used in non-standard analysis
* Hyperreal.org, a rave culture website based in San Francisco, US
* Hyperreality, a term used in semiotics and po ...
approach, the quantity
is taken to be an infinitesimal, a nonzero number that is closer to 0 than to any nonzero real. The manipulations displayed above then show that
is infinitely close to 2''x'', so the derivative of ''f'' at ''x'' is then 2''x''.
Discarding the "error term" is accomplished by an application of the
standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
. Dispensing with infinitesimal error terms was historically considered paradoxical by some writers, most notably
George Berkeley
George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immate ...
.
Once the hyperreal number system (an infinitesimal-enriched continuum) is in place, one has successfully incorporated a large part of the technical difficulties at the foundational level. Thus, the
epsilon, delta techniques that some believe to be the essence of analysis can be implemented once and for all at the foundational level, and the students needn't be "dressed to perform multiple-quantifier logical stunts on pretense of being taught
infinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
", to quote a recent study. More specifically, the basic concepts of calculus such as continuity, derivative, and integral can be defined using infinitesimals without reference to epsilon, delta (see next section).
Keisler's textbook
Keisler's
Elementary Calculus: An Infinitesimal Approach defines continuity on page 125 in terms of infinitesimals, to the exclusion of epsilon, delta methods.
The derivative is defined on page 45 using infinitesimals rather than an epsilon-delta approach.
The integral is defined on page 183 in terms of infinitesimals.
Epsilon, delta definitions are introduced on page 282.
Definition of derivative
The
hyperreals
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
can be constructed in the framework of
Zermelo–Fraenkel set theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as ...
, the standard axiomatisation of set theory used elsewhere in mathematics. To give an intuitive idea for the hyperreal approach, note that, naively speaking, nonstandard analysis postulates the existence of positive numbers ε ''which are infinitely small'', meaning that ε is smaller than any standard positive real, yet greater than zero. Every real number ''x'' is surrounded by an infinitesimal "cloud" of hyperreal numbers infinitely close to it. To define the derivative of ''f'' at a standard real number ''x'' in this approach, one no longer needs an infinite limiting process as in standard calculus. Instead, one sets
:
where st is the
standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
, yielding the real number infinitely close to the hyperreal argument of st, and
is the natural extension of
to the hyperreals.
Continuity
A real function ''f'' is continuous at a standard real number ''x'' if for every hyperreal ''x' '' infinitely close to ''x'', the value ''f''(''x' '') is also infinitely close to ''f''(''x''). This captures
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
's definition of continuity as presented in his 1821 textbook
Cours d'Analyse
''Cours d'Analyse de l’École Royale Polytechnique; I.re Partie. Analyse algébrique'' is a seminal textbook in infinitesimal calculus published by Augustin-Louis Cauchy in 1821. The article follows the translation by Bradley and Sandifer in ...
, p. 34.
Here to be precise, ''f'' would have to be replaced by its natural hyperreal extension usually denoted ''f''
* (see discussion of
Transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...
in main article at
nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
).
Using the notation
for the relation of being infinitely close as above,
the definition can be extended to arbitrary (standard or nonstandard) points as follows:
A function ''f'' is ''
microcontinuous'' at ''x'' if whenever
, one has
Here the point x' is assumed to be in the domain of (the natural extension of) ''f''.
The above requires fewer quantifiers than the
(''ε'', ''δ'')-definition familiar from standard elementary calculus:
''f'' is continuous at ''x'' if for every ''ε'' > 0, there exists a ''δ'' > 0 such that for every ''x' '', whenever , ''x'' − ''x' '', < ''δ'', one has , ''f''(''x'') − ''f''(''x' ''), < ''ε''.
Uniform continuity
A function ''f'' on an interval ''I'' is
uniformly continuous
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
if its natural extension ''f''* in ''I''* has the following property (see Keisler, Foundations of Infinitesimal Calculus ('07), p. 45):
for every pair of hyperreals ''x'' and ''y'' in ''I''*, if
then
.
In terms of microcontinuity defined in the previous section, this can be stated as follows: a real function is uniformly continuous if its natural extension f* is microcontinuous at every point of the domain of f*.
This definition has a reduced quantifier complexity when compared with the standard
(ε, δ)-definition. Namely, the epsilon-delta definition of uniform continuity requires four quantifiers, while the infinitesimal definition requires only two quantifiers. It has the same quantifier complexity as the definition of uniform continuity in terms of ''sequences'' in standard calculus, which however is not expressible in the
first-order language
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
of the real numbers.
The hyperreal definition can be illustrated by the following three examples.
Example 1: a function ''f'' is uniformly continuous on the semi-open interval (0,1], if and only if its natural extension f* is microcontinuous (in the sense of the formula above) at every positive infinitesimal, in addition to continuity at the standard points of the interval.
Example 2: a function ''f'' is uniformly continuous on the semi-open interval [0,∞) if and only if it is continuous at the standard points of the interval, and in addition, the natural extension ''f''* is microcontinuous at every positive infinite hyperreal point.
Example 3: similarly, the failure of uniform continuity for the squaring function
:
is due to the absence of microcontinuity at a single infinite hyperreal point, see below.
Concerning quantifier complexity, the following remarks were made by Kevin Houston (mathematician), Kevin Houston:
:The number of quantifiers in a mathematical statement gives a rough measure of the statement’s complexity. Statements involving three or more quantifiers can be difficult to understand. This is the main reason why it is hard to understand the rigorous definitions of limit, convergence, continuity and differentiability in analysis as they have many quantifiers. In fact, it is the alternation of the
and
that causes the complexity.
Andreas Blass
Andreas Raphael Blass (born October 27, 1947) is a mathematician, currently a professor at the University of Michigan. He works in mathematical logic, particularly set theory, and theoretical computer science.
Blass graduated from the University ...
wrote as follows:
:Often ... the nonstandard definition of a concept is simpler than the standard definition (both intuitively simpler and simpler in a technical sense, such as quantifiers over lower types or fewer alternations of quantifiers).
[, p. 37.]
Compactness
A set A is compact if and only if its natural extension A* has the following property: every point in A* is infinitely close to a point of A. Thus, the open interval (0,1) is not compact because its natural extension contains positive infinitesimals which are not infinitely close to any positive real number.
Heine–Cantor theorem
The fact that a continuous function on a compact interval ''I'' is necessarily uniformly continuous (the
Heine–Cantor theorem
In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if f \colon M \to N is a continuous function between two metric spaces M and N, and M is compact, then f is uniformly continuous. An important speci ...
) admits a succinct hyperreal proof. Let ''x'', ''y'' be hyperreals in the natural extension ''I*'' of ''I''. Since ''I'' is compact, both st(''x'') and st(''y'') belong to ''I''. If ''x'' and ''y'' were infinitely close, then by the triangle inequality, they would have the same standard part
:
Since the function is assumed continuous at c,
:
and therefore ''f''(''x'') and ''f''(''y'') are infinitely close, proving uniform continuity of ''f''.
Why is the squaring function not uniformly continuous?
Let ''f''(''x'') = ''x''
2 defined on
. Let
be an infinite hyperreal. The hyperreal number
is infinitely close to ''N''. Meanwhile, the difference
:
is not infinitesimal. Therefore, ''f*'' fails to be microcontinuous at the hyperreal point ''N''. Thus, the squaring function is not uniformly continuous, according to the definition in
uniform continuity
In mathematics, a real function f of real numbers is said to be uniformly continuous if there is a positive real number \delta such that function values over any function domain interval of the size \delta are as close to each other as we want. In ...
above.
A similar proof may be given in the standard setting .
Example: Dirichlet function
Consider the
Dirichlet function
In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number).
\mathbf 1_\Q(x) = \begin
1 ...
:
It is well known that, under the
standard definition of continuity, the function is discontinuous at every point. Let us check this in terms of the hyperreal definition of continuity above, for instance let us show that the Dirichlet function is not continuous at π. Consider the continued fraction approximation a
n of π. Now let the index n be an infinite
hypernatural
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
number. By the
transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...
, the natural extension of the Dirichlet function takes the value 1 at a
n. Note that the hyperrational point a
n is infinitely close to π. Thus the natural extension of the Dirichlet function takes different values (0 and 1) at these two infinitely close points, and therefore the Dirichlet function is not continuous at ''π''.
Limit
While the thrust of Robinson's approach is that one can dispense with the approach using multiple quantifiers, the notion of limit can be easily recaptured in terms of the
standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
st, namely
:
if and only if whenever the difference ''x'' − ''a'' is infinitesimal, the difference ''f''(''x'') − ''L'' is infinitesimal, as well, or in formulas:
:if st(''x'') = ''a'' then st(''f''(''x'')) = L,
cf.
(ε, δ)-definition of limit
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...
.
Limit of sequence
Given a sequence of real numbers
, if
''L'' is ''the limit'' of the sequence and
:
if for every infinite
hypernatural
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is ...
''n'', st(''x''
''n'')=''L'' (here the extension principle is used to define ''x''
''n'' for every hyperinteger ''n'').
This definition has no
quantifier alternations. The standard
(ε, δ)-style definition, on the other hand, does have quantifier alternations:
:
Extreme value theorem
To show that a real continuous function ''f'' on
,1has a maximum, let ''N'' be an infinite
hyperinteger
In nonstandard analysis, a hyperinteger ''n'' is a hyperreal number that is equal to its own integer part. A hyperinteger may be either finite or infinite. A finite hyperinteger is an ordinary integer. An example of an infinite hyperinteger is g ...
. The interval
, 1has a natural hyperreal extension. The function ''f'' is also naturally extended to hyperreals between 0 and 1. Consider the partition of the hyperreal interval
,1into ''N'' subintervals of equal
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
length 1/''N'', with partition points ''x''
''i'' = ''i'' /''N'' as ''i'' "runs" from 0 to ''N''. In the standard setting (when ''N'' is finite), a point with the maximal value of ''f'' can always be chosen among the ''N''+1 points ''x''
''i'', by induction. Hence, by the
transfer principle
In model theory, a transfer principle states that all statements of some language that are true for some structure are true for another structure. One of the first examples was the Lefschetz principle, which states that any sentence in the first- ...
, there is a hyperinteger ''i''
0 such that 0 ≤ ''i''
0 ≤ ''N'' and
for all ''i'' = 0, …, ''N'' (an alternative explanation is that every
hyperfinite set
In nonstandard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set ''H'' of internal cardinality ''g'' ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal b ...
admits a maximum). Consider the real point
:
where st is the
standard part function
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
. An arbitrary real point ''x'' lies in a suitable sub-interval of the partition, namely