A common task in mathematics is to prove that a given sequence converges. One idea that sometimes helps is to pass to a subsequence of that has better properties than the original sequence. This article discusses various implementations of this idea.
Suppose we are able to prove that is a sequence in a totally bounded set in a complete metric space. Then it is natural to look at the subsequences of because we know that any subsequence of has a convergent subsubsequence. If for every subsequence we can find a subsubsequence converging to the same function then we know that converges to as well. This method is explained in the article To prove that a sequence converges, find one or more convergent subsequences.
Another way to use subsequences to prove convergence is to pick a particular subsequence whose rapidity of convergence is relatively easy to derive and understand and which we expect to approximate the whole sequence well. Once it is shown that this subsequence converges one proves that the rest of the sequence follows this subsequence closely. This method is explained in the article To prove convergence, find a rapidly converging subsequence.