Every convergent sequence is bounded.


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 \((a_n)\) be a sequence that convergences to some number \(l\). By definition (2.2.3), we know that there exists an \(N \in \mathbf{N}\) such that if \(n \geq N\), then for any \(\epsilon > 0\), we have \(|a_n - l| \leq \epsilon\). If we let \(\epsilon = 1\), then we can conclude that \(a_n\) is in the interval \((l - 1, l + 1)\). Since we don’t know if \(l\) if positive or negative, we can just use \(|l|+1\) (why?) and conclude

$$ \begin{align*} |x_n| < |l| + 1 \text{whenever $n \geq N$}. \end{align*} $$

For the terms that come before \(N\), let \(M = \max\{|x_1|,|x_2|,...,|x_{N-1}|, |l|+1\}\) and pick \(M\) to be the new bound such that

$$ \begin{align*} |x_n| < M \quad \text{whenever $n \in \mathbf{N}$}. \end{align*} $$

\(\blacksquare\)

References: