Monotonic Sequence Convergence Theorem, Part 1: A bounded, monotonic increasing sequence is convergent.
Proof: Let be a monotonic increasing sequence, . is a bounded set of real numbers , thus, via the completeness axiom of real numbers, it must have a supremum. We call the supremum . Thus for all . We want to show that
Let . . For and . Thus, for all , which implies that converges to .