Monotone Convergence Theorem

theoremreal-analysis

Intuition

Formal Statement

Any bounded and monotone sequence is convergent.

Proof

We can assume $(x_n{)}_n$ is increasing (note that if a sequence $(y_n{)}_n$ is decreasing, then $(-y_n{)}_n$ is increasing).

We know that $\left|x_n\right|\le m$ for all $n$, and that $x_n\le x_{n+1}$ for all $n$. We want to show there exists $L\in \mathbb{R}$ such that ${\lim }_{n\to \infty }{x}_n=L$.

Let $S=\left\{x_n\mid x_n\in (x_n{)}_n\right\}$. Since $S$ is bounded, it has a supremum $\sup S$…

Corollaries