**Lemma**: Any [[Real Cauchy Sequence]] is [[Bounded Sequence|bounded]]. **Proof**: Again, take $\epsilon = 1$. By definition of Cauchy sequence $\exists N \text{ s.t } |a_{n}-a_{N}|< 1 $So using [[Absolute Value Satisfies Triangle Inequality|triangle inequality]]$|a_{n}| = |a_{n}-a_{N}+a_{N}| \leq |a_{n}-a_{N}|+|a_{N}|<1+|a|$Define $A = max(|a_{1}|,|a_{2}|, |a_{3}|,\dots|a_{N-1}|,1+|a_{N}|)$then $a_{n}<A$ for all $n \in \mathbb{N}$ so $(a_{n})$ is bounded. ### References - [[General Principle of Convergence]]