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Prove that a cauchy sequence w/ a convergent subsequence is itself convergent. The subquence is convergent, so the following holds: $$\forall \epsilon > 0, \exists M \in \mathbb{N}\ |\ a \geq M \implies |x-x_a| < \epsilon$$ And for our parent cauchy sequence, we have $$\forall \delta > 0, \exists N \in \mathbb{N} \ | \ n,m \geq N |x_n - x_m|$$ So choose $|x_n - x_m| > x_a$, so that $|x - |x_n - x_m|| < |x - x_a| < \epsilon$. Thus a cauchy sequence containing a convergent subsequence is itself convergent.
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Wed, 20 Oct 2021 22:28 GMT