## Proof of the Week for 20120730

Monotonic Sequence Convergence Theorem, Part 1: A bounded, monotonic increasing sequence is convergent.

Proof: Let $\left(a_n\right)$ be a monotonic increasing sequence, $a_n\in\mathbb{R}$. $\left(a_n\right)$ is a bounded set of real numbers , thus, via the completeness axiom of real numbers, it must have a supremum. We call the supremum $A$. Thus $a_n\le A$ for all $n \in\mathbb{N}$. We want to show that $\lim_{n \to \infty}a_n = A$

Let $\epsilon > 0$. $A - \epsilon < a_m$. For $n \ge m, a_n \ge a_m$ and $A - \epsilon < a_n \le A$. Thus, $0 \le A - a_n < \epsilon$ for all $n \ge m$, which implies that $\left(a_n\right)$ converges to $A$.