The Limit You Never Have to Compute

Bounded wasn't enough — a bounded sequence can oscillate forever and never settle. Add one hypothesis, "never turn around," and something remarkable happens: a limit is guaranteed to exist. You don't have to find it, guess it, or compute it. This video covers monotone sequences, the monotone convergence theorem, and why completeness is the axiom doing all the work. Plus the trap that catches almost everyone: proving boundedness by looking at a graph. 00:00 Intro 00:05 What monotone buys you 00:32 Increasing, decreasing, eventually 01:07 The monotone convergence theorem 01:36 Why completeness is the engine 02:11 Bounded is where the work is 02:47 Proving it properly 03:32 Existence first, then value 04:11 No third option You'll want sequence limits and boundedness first. Mathique — ideas, actually explained.