A standard theorem in analysis says that if $\{x_n\}$ is a Cauchy sequence in a normed space $X$, then $\{x_n\} \rightarrow x$ if and only if it has a convergent subsequence $\{x_{n_k}\}$ such that $\{x_{n_k}\} \rightarrow x$.
What are some examples that show the usefulness of this result? I am able to prove it, but I don't see how it simplifies proofs. For example, in their book on functional analysis, Reed & Simon give a hint to one of their exercises that says
To show that a Cauchy sequence converges it is only necessary to show that a subsequence converges.