Monotone Convergence Theorem (Real Analysis)

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This proof is about convergence in real analysis. For other uses, see Monotone Convergence Theorem.

Theorem

Every bounded monotone sequence is convergent.


Let $\sequence {x_n}$ be a sequence in $\R$.


Increasing Sequence

Let $\sequence {x_n}$ be increasing and bounded above.


Then $\sequence {x_n}$ converges to its supremum.


Decreasing Sequence

Let $\sequence {x_n}$ be decreasing and bounded below.


Then $\sequence {x_n}$ converges to its infimum.


Graphical Illustration

The following diagram illustrates the Monotone Convergence Theorem:


Monotone-convergence-theorem.png


Examples

Example: $\dfrac {n - 1} n$

The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:

$a_n = \dfrac {n - 1} n$

is convergent to the limit $1$.


Example: $x^n$ for $0 < x < 1$

Let $x \in \R$ such that $0 < x < 1$.

The sequence $\sequence {a_n}_{n \mathop \ge 1}$ defined as:

$a_n = x^n$

is convergent to the limit $0$.


Sources