A very widely stated result:

A sequence $x\_n \to x$ iff every subsequence $x\_{n^{\prime}}$ of $x\_n$ contains a further subsequence $x\_{n^{\prime\prime}}$ such that $x\_{n^{\prime\prime}}\to x$.

My question is that there is no need to pass on to sub-sub sequence. The statement is true for sub sequence replacing sub-sub sequence. Then why is this statement useful?? Is it the case that in several places its difficult to find a convergent sub-sequence but easy to get a convergent sub-sub sequence??