How can I prove this proposition?

Let $(a_n)$ be a sequence. If $(|a_n|)$ is convergent then $(a_n)$ has a convergent subsequence.

If $(|a_n|)$ converges to $L$ then $\forall \epsilon>0, \exists n_0 \in \mathbb{N}$ where $n>n_0 \implies ||a_n|-L|<\epsilon$.

I got stuck here.


2 Answers 2



There are either infinitely many non-negative elements of $a_n$ or infinitely many non-positive elements. Consider the subsequence consisting of these.


Use Bolzano Weierstrass theorem. Every bounded sequence have subsequence convergence.


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