Cauchy Sequence is Bounded/Real Numbers/Proof 1

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 $m, n \ge N$.

In particular, by the Triangle Inequality, for all $m \ge N$:

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