Let $\{a_n\}$ be a Cauchy Sequence in $M$. Prove that the set $\{a_n\}$ is bounded.

Question

Let $\{a_n\}$ be a Cauchy Sequence in $M$. Prove that the set $\{a_n\}$ is bounded.

Proof: let $\{a_n\}$ be a Cauchy sequence in $M$, then there exists a $N \in \Bbb N$ such that $n,m \geq N$ implies $d(a_n,a_m) < \varepsilon$.

Thus, $a_n$ is bounded by $N$.

Is this correct?

Answer 1

Hint:

For $\epsilon = 1$ there exists an $N$ such that for all $n, m > N$, $|a_n - a_m| < 1$ and in particular $|a_{N+1} - a_m| < 1$ for all $m > N$.

Can you now find an upper and lower bound on all $a_n$?