Subsequence Convergence Theorems

Real Analysis · Axiom Academy

LESSON Subsequences Inherit the Limit If , then every subsequence — and that one fact tells you exactly when a sequence must diverge. 1. Every Subsequence Is Pulled to the Same Limit Suppose the whole sequence (a_n) converges to L . A subsequence (a_ n_k ) just reads off some of those terms at indices . Watch the full sequence settle into the band around L , then watch three different subsequences get extracted — each one lands in the same band . 2. Why It Works — and the Test It Gives You The engine is a tiny inequality. Because the indices strictly increase through the naturals, the k -th one can never lag behind k : . So once k is past the convergence cutoff N , the index n_k is past N too, and that term already lives inside the -window. Watch each subsequence index get dragged up to or beyond its own height k . 3. The Classic Counterexample: a_n = (-1)^n Now use the test in reverse. The sequence never settles. Split it by parity: the even-index subsequence is all +1 , the odd-index subsequence is all -1 . Two subsequences, two different limits — watch them separate. One fact, used both directions: convergence flows down to every subsequence, and disagreeing subsequences flow back up to divergence. Scroll up to replay any animation.

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