Cauchy subsequence
= Cauchy subsequence
{c}
A Cauchy subsequence is a <subsequence> that is a <Cauchy sequence>. Every <subsequence> of a <Cauchy sequence> is a <Cauchy sequence>, because its increasing indices eventually lie beyond any index required by the original pairwise bound. Every <convergent subsequence> is also a <Cauchy sequence>.
The converse about the original <sequence> fails: $a_n=(-1)^n$ has a constant <subsequence> at its <even integers>, hence a <Cauchy sequence>, while $a_n$ itself is not a <Cauchy sequence>. If the original <sequence> is already a <Cauchy sequence> and one <subsequence> converges to $L$, however, the whole <sequence> converges to $L$. Choose a late subsequence term close to $L$ and use the <triangle inequality> to compare it with every sufficiently late original term. https://www.ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/mit18_100af20_lec102.pdf[MIT 18.100A, Lecture 10, Theorem 6: a convergent subsequence of a Cauchy sequence]{ref}.