Sequence Converges iff Every Subsequences Converge to the Same Limit | Real Analysis

A sequence converges to a limit L if and only if every subsequence converges to L. We prove this wonderful result about subsequences in real analysis in today's video lesson! First we prove that if a sequence converges to a limit, then all of its subsequences converge to that same limit. To do this, we need only consider an arbitrary subsequence, and use the fact that the original sequence converges to finish things up. Then we need to prove if every subsequence converges to the same limit that the original sequence does as well. This is trivial because every sequence is a subsequence of itself.
Intro to Subsequences: • Intro to Subsequences ...
#realanalysis #math
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