Monotone Convergence Theorem (Real Analysis)

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



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 $0$.