Cauchy Sequence is Bounded/Real Numbers

Theorem
Every Cauchy sequence in $\R$ is bounded.

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 $m, n \ge N$.

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

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