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 .

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