Let \(\{a_n\}\) be a sequence. Prove that if \(\lim\limits_{n \rightarrow \infty} |a_n| = 0\), then \(\lim\limits_{n \rightarrow \infty} a_n = 0\).
Definitions of sequences and convergence: here, Definitions of subsequences: here.
For the “show the limit” template: here.
Proof
We are given that \(\lim\limits_{n \rightarrow \infty} |a_n| = 0\). By definition, this means that for any \(\epsilon > 0\), there exists an \(N \in \mathbb{N}\) such that for all \(n \geq N\), we must have
$$
\begin{align*}
\left| |a_n| - 0 \right| &< \epsilon
\end{align*}
$$
But \(| |a_n| - 0 | = |a_n|\). Therefore
$$
\begin{align*}
\left| a_n - 0 \right| &< \epsilon
\end{align*}
$$
This is true for all \(\epsilon > 0\) and all \(n \geq N\). But this is exactly the definition of convergence for a sequence. Therefore, we must have \(\lim\limits_{n \rightarrow \infty} a_n = 0\) as we wanted to show. \(\blacksquare\)
References
- Problem Statement Source: Aleph 0