Cauchy Sequence is Bounded/Real Numbers/Proof 2

Proof
Let $\sequence {a_n}$ be a Cauchy sequence in $\R$.

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

for all $m, n \ge N$.

Note that for $m \le N$:

Hence for all $m \in \N$:


 * $\size {a_m} < \max \set {\size {a_1}, \size {a_2}, \dotsc, \size {a_N} } + 1$

Therefore $\sequence {a_n}$ is bounded, as claimed.