Bounded Sequences
Definition: Bounded Sequences
A sequence \(x_n\) is bounded if there exists a number \(M > 0\) such that every term in the sequence \(|x_n| \leq M\) for all \(n \in \mathbf{N}\).
Proof:
Assume \((x_n)\) converges to some limit \(l\). This means that for any \(\epsilon\), there must exist some natural number \(N\) such that if \(n \geq N\), then \(x_n\) will be in the interval \((l -\epsilon, l + \epsilon)\). Since we’re searching for any bound, we can set \(\epsilon\) to any value like 1. Also, whether \(l\) is positive or negative, we can conclude that for all \(n \geq N\) that
$$
\begin{align*}
|x_n| \leq |l| + 1
\end{align*}
$$
holds. Since this bound only applies to the terms when \(n \geq N\), we can take the maximum of all the early terms to up to \(N\) and \(|l|+1\). So,
$$
\begin{align*}
M = \max{\{|x_1|, |x_2|, |x_3|, ....,|x_{N+1}, |l| + 1\}}.
\end{align*}
$$
Therefore, we have \(|x_n| \leq M\) for all \(n \in \mathbf{N}\) as required. \(\blacksquare\).