If two subsequences of a sequence converge to different limits or if a subsequence diverge then the sequence diverges
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 and let two subsequences converge to different limits or have a subsequence diverge. Suppose for the sake of contradiction that \((a_n)\) converges to a limit \(a\). This means that every subsequence of \((a_{n_k})\) must converge \(a\) by the proof we did earlier here. But this is a contradiction since we assumed that two subsequences converge to different limits or that a subsequence diverges. Therefore, \((a_n)\) must diverge as required. \(\blacksquare\)
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