Monotonic and Bounded Sequences

Calculus 2 · Axiom Academy

LESSON Monotonic and Bounded Sequences When does a sequence have to converge — even before you find its limit? Two properties together force the answer. Increasing: each term is at least the previous one: for all n Decreasing: each term is at most the previous one: for all n Think of a monotonic sequence as walking consistently uphill or downhill — you never change direction. This does not mean the sequence reaches a destination (converges), but it means the journey is predictable. Bounded above: there exists M with for all n Bounded below: there exists m with for all n Bounded: both bounded above and below A bounded sequence is like a ball bouncing inside a box — it can move around freely, but it never escapes certain walls. It may oscillate wildly, yet every term stays between the bounds. 3. The Monotone Convergence Theorem This is one of the most powerful results in analysis. If you walk consistently uphill (increasing) but there is a ceiling (an upper bound), you must eventually approach some limiting height. Likewise, walking downhill (decreasing) above a floor (a lower bound) forces you toward some limiting depth. Watch an increasing sequence pressed under its ceiling. Each term is forced above the last yet kept below M , so the gap to the ceiling can only shrink — the terms have nowhere to go but onto a limit L . 4. Why It Works: Nested Intervals

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