monotone convergence theorem
This theorem says that a sequence which only climbs but is held under a ceiling has nowhere to go but up to that ceiling's lowest possible value. Intuitively, an increasing sequence that cannot escape upward must pile up against its tightest upper barrier — and that pile-up point is its limit.
Precisely: every increasing sequence that is bounded above converges, and its limit equals the supremum of its set of values; dually, every decreasing sequence that is bounded below converges to the infimum of its values. So for monotone sequences, bounded is exactly equivalent to convergent — a rare clean 'if and only if' in the theory of limits.
The theorem is a direct consequence of the completeness of the real numbers: it is the supremum's existence (guaranteed by the least upper bound property) that supplies the limit. Indeed, over the rationals the statement is false, because a bounded increasing sequence of rationals can have its supremum sitting at an irrational number not present in the rationals. This is one of the cleanest illustrations of why completeness matters.
Define a_1 = 1 and a_{n+1} = (a_n + 2/a_n)/2. One checks it is decreasing and bounded below by sqrt(2), so by the theorem it converges; passing to the limit L = (L + 2/L)/2 gives L = sqrt(2). This is the ancient Babylonian method for square roots.
Monotone convergence both guarantees the limit exists and lets you solve for it.
The theorem gives existence of the limit and identifies it as a sup or inf, but it does not tell you how fast the sequence converges, nor a closed form for the limit.