# lemma: Cauchy sequences are bounded.

Lemma: Cauchy sequences are bounded.

Proof: Given Cauchy sequence $(s_n)$, for $\varepsilon=1$ we obtain $N\in\Bbb N$ such that $m, n > N$ implies $|s_n − s_m| < 1$.

(2) implies $|s_n − s_{N+1}| < 1$ for all $n\geq N$, (from this step (2) to the next step (3) how does one get there? is it through the reverse triangle inequality ? do we simply say that $||s_n|-|s_{N+1}||\leq|s_n-s_{N+1}|<1$

then $||s_n|-|s_{N+1}|| < 1$ for all $n\geq N$

then add $|s_{N+1}|$ to both sides of the inequality? is it that simple?)

(3) implies $|s_n| < |s_{N+1}| + 1$ for all $n>N$. If $M = \max\{|s_{N+1}| + 1, |s_1|, |s_2|,\ldots, |s_N |\}$, then $|s_n| \leq M$ for all $n \in\Bbb N$ hence the sequence $(s_n)$ is bounded. QDE

4) QED