If \(\{a_n\}\) is a convergent sequence with \(\lim\limits_{n \rightarrow \infty} a_n = L\). Prove that \(\{ |a_n|\}\) converges as well. Does \(\{ |a_n|\}\) converges to \(L\)?

Definitions of sequences and convergence: here, Definitions of subsequences: here.
For the “show the limit” template: here.


Proof

Since \(\lim a_n = L\), then by definition this means that for all \(\epsilon > 0\), there exists some \(N \in \mathbb{N}\) such that for all \(n \geq N\), we must have

$$ \begin{align*} \big\lvert a_n - L \big\rvert &< \epsilon \end{align*} $$

By the reverse triangle inequality,

$$ \begin{align*} \big\lvert |a_n| - |L| \big\rvert &\leq \big\lvert a_n - L \big\rvert \\ &< \epsilon \end{align*} $$

Since this is true for all \(\epsilon > 0\) such that \(n \geq N\). Then, we can conclude that

$$ \begin{align*} \lim |a_n| = |L| \end{align*} $$

Therefore, the \(\{|a_n|\}\) converges to \(|L|\). \(\blacksquare\)


References