Cauchy Sequence is Bounded/Real Numbers/Proof 2

Proof
Let $\left \langle {a_n} \right \rangle$ be a Cauchy sequence in $\R$.

Then there exists $N \in \N$ such that:
 * $\left\vert{a_m - a_n}\right\vert < 1$

for all $m, n \ge N$.

Note that for $m \le N$:

Hence for all $m \in \N$:


 * $\left\vert{a_m}\right\vert < \max \left\{ {\left\vert{a_1}\right\vert, \left\vert{a_2}\right\vert, \cdots, \left\vert{a_N}\right\vert}\right\} + 1$

Therefore $\left\langle{a_n}\right\rangle$ is bounded, as claimed.