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
- Problem Statement Source: Aleph 0