Convergence Detective

Intro to Proofs · Axiom Academy

A sequence converges when its terms eventually settle near one value — and stay there. Watch the terms hop in, then put the rule in your hands. What does it actually mean for a sequence to converge? "The terms get close to L " is the right picture — but close how, and close to stay ? The whole subject of analysis turns on pinning that down. Here is the idea you can see: draw a thin band around a target value, and a convergent sequence will eventually drop every remaining term into that band and never leave. Watch the terms of get plotted left to right. The shaded band is everything within of the limit L = 0 . The first few terms land outside it — but past some index N , every term lands inside and stays inside. Convergence is not "the terms get small" — it is "past some point, the terms never leave the band, no matter how the rest of the sequence wanders before then." Shrink the band as tight as you like — the tail still fits Drag smaller. The band around L = 0 tightens, and the cutoff N — the first index past which all the terms of stay inside — slides to the right. The point: no matter how mean you are with , such an N always exists. That is exactly what " " means. Smaller needs a larger N — here — but a finite N always works. Infinitely many terms still get trapped; only finitely many ever escape. Converge or diverge? Look at the tail

This is the written version of the interactive lesson above. See the full Intro to Proofs course.