Monotone Convergence Theorem

Real Analysis · Axiom Academy

LESSON The Monotone Convergence Theorem A sequence that only climbs but stays under a ceiling has nowhere to go but its least upper bound: . 1. The Ceiling Has to Catch It Picture an increasing sequence: every new term sits at or above the last, so the points only ever march upward . Now hang a ceiling M above them — an upper bound the terms may never cross. The terms keep rising, yet they can never break through M . Trapped between "always going up" and "can't pass M ," they have only one option: pile up against some height and settle there. monotone increasing and bounded above the limit is exactly the least upper bound If (a_n) is increasing and bounded above, then . If (a_n) is decreasing and bounded below, then . 2. Why It Lands on the Supremum To prove convergence to , pick any tolerance and look at the line . Because L is the least upper bound, is too low to be an upper bound — so some term a_N has already climbed above it. Now monotonicity finishes the job: every later term is at least a_N , and none can exceed L . So from N onward, the whole tail is squeezed into the strip between and L . 3. Seeing It on a Real Sequence Take the recursively defined sequence . Each step feeds the last term back in. Watch the staircase : every term lands higher than the one before (increasing), yet every term stays below the ceiling M=2 (bounded above). By the theorem it must converge — and the staircase climbs straight into the corner where . The first few terms climb and crowd

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