In the mathematical field of
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include co ...
, the monotone convergence theorem is any of a number of related theorems proving the good
convergence
Convergence may refer to:
Arts and media Literature
*''Convergence'' (book series), edited by Ruth Nanda Anshen
*Convergence (comics), "Convergence" (comics), two separate story lines published by DC Comics:
**A four-part crossover storyline that ...
behaviour of
monotonic sequences, i.e. sequences that are non-
increasing, or non-
decreasing. In its simplest form, it says that a non-decreasing
bounded-above sequence of real numbers
converges to its smallest upper bound, its
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its
infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded.
For sums of non-negative increasing sequences
, it says that taking the sum and the supremum can be interchanged.
In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in
measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
due to
Lebesgue and
Beppo Levi that says that for sequences of non-negative pointwise-increasing
measurable function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s
, taking the integral and the supremum can be interchanged with the result being finite if either one is finite.
Convergence of a monotone sequence of real numbers
Every bounded-above monotonically nondecreasing sequence of real numbers is convergent in the real numbers because the supremum exists and is a real number. The proposition does not apply to rational numbers because the supremum of a sequence of rational numbers may be irrational.
Proposition
(A) For a non-decreasing and bounded-above sequence of real numbers
:
the limit
exists and equals its
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
:
:
(B) For a non-increasing and bounded-below sequence of real numbers
:
the limit
exists and equals its
infimum:
:
.
Proof
Let
be the set of values of
. By assumption,
is non-empty and bounded above by
. By the
least-upper-bound property of real numbers,
exists and
. Now, for every
, there exists
such that
, since otherwise
is a strictly smaller upper bound of
, contradicting the definition of the supremum
. Then since
is non decreasing, and
is an upper bound, for every
, we have
:
Hence, by definition
.
The proof of the (B) part is analogous or follows from (A) by considering
.
Theorem
If
is a monotone
sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
of
real number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, i.e., if
for every
or
for every
, then this sequence has a finite limit
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
the sequence is
bounded.
Proof
* "If"-direction: The proof follows directly from the proposition.
* "Only If"-direction: By
(ε, δ)-definition of limit
Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1.
In mathematics, the limit of a function is a fundame ...
, every sequence
with a finite limit
is necessarily bounded.
Convergence of a monotone series
There is a variant of the proposition above where we allow unbounded sequences in the extended real numbers, the real numbers with
and
added.
:
In the extended real numbers every set has a
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
(resp.
infimum) which of course may be
(resp.
) if the set is unbounded. An important use of the extended reals is that any set of non negative numbers
has a well defined summation order independent sum
:
where
are the upper extended non negative real numbers. For a series of non negative numbers
:
so this sum coincides with the sum of a series if both are defined. In particular the sum of a series of non negative numbers does not depend on the order of summation.
Monotone convergence of non negative sums
Let
be a sequence of non-negative real numbers indexed by natural numbers
and
. Suppose that
for all
. Then
:
Proof
Since
we have
so
.
Conversely, we can interchange sup and sum for finite sums by reverting to the
limit definition, so
hence
.
Examples
Matrices
The theorem states that if you have an infinite matrix of non-negative real numbers
such that the rows are weakly increasing and each is bounded
where the bounds are summable
then, for each column, the non decreasing column sums
are bounded hence convergent, and the limit of the column sums is equal to the sum of the "limit column"
which element wise is the supremum over the row.
''e''
Consider the expansion
:
Now set
:
for
and
for
, then
with
and
:
.
The right hand side is a non decreasing sequence in
, therefore
:
.
Beppo Levi's lemma
The following result is a generalisation of the monotone convergence of non negative sums theorem above to the measure theoretic setting. It is a cornerstone of measure and integration theory with many applications and has
Fatou's lemma and the
dominated convergence theorem as direct consequence. It is due to
Beppo Levi, who proved a slight generalization in 1906 of an earlier result by
Henri Lebesgue.
[
]
Let
denotes the
-algebra of Borel sets on the upper extended non negative real numbers