Monotone Convergence Theorem (Real Analysis)

This article was Featured Proof between 30 June 2009 and 6 July 2009.

This proof is about Monotone Convergence Theorem in the context of Real Analysis. For other uses, see Monotone Convergence Theorem.

Theorem

Then $\sequence {x_n}$ is convergent.

Let $\sequence {x_n}$ be an increasing real sequence which is bounded above.

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

Let $\sequence {x_n}$ be a decreasing real sequence which is bounded below.

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

The following diagram illustrates the Monotone Convergence Theorem:

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

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

is convergent to the limit $1$.

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 $0$.