Prove that \((a_n) = (-1)^n\) 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 \((a_n) = (-1)^n\). Consider the subsequence \((a_{2n})\). This is a subsequence of \((a_n)\). \(a_{2n}\) is

$$ \begin{align*} \{1,1,1,1,1,1,1,1,...\} \end{align*} $$

This subsequence converges to 1. Consider now the subsequence \((a_{2n}) = \{-1,-1,-1,-1,....\}\). This subsequence converges to -1. Since \((a_n)\) has two subsequences that converge to two different limits than \((a_n)\) must diverge due to this. \(\blacksquare\)

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