 Cauchy subsequence

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. ^[ref].