A sequence \(a_n\) converges to \(a\) if and only if every subsequence of \(a_n\) also converges to \(a\).


For the absolute value function definition and other properties see here.
For the definitions of sequences and what it means to for a sequence to converge, see this, for subsequences see this.
For the “show the limit” template and an example, see this.

Proof:

Let \((a_n)\) be a sequence. The statement, if a sequence \(a_n\) converges to \(a\) then every subsequence also converges to \(a\), is proved in here. We now prove the other direction that claims that if every subsequence converges to \(a\) then \((a_n)\) also converges to \(a\). Assume that every subsequence of \((a_n)\) converges to \(a\). But \((a_n)\) is also a subsequence of itself. Take \((a_{n_k})\) and set \(n_k = k\). so \(n_1\) is the first term of the sequence, \(n_2\) is the second term of the sequence and so on. Therefore, \((a_n)\) also converges to \(a\). \(\blacksquare\)

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