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 $m, n \ge N$.

Note that
 * $\left\vert{a_m}\right\vert < \max\{\left\vert{a_1}\right\vert,\left\vert{a_2}\right\vert,\cdots,\left\vert{a_N}\right\vert\}+1$ for all $m \in \N$.

Since for $m \le N$

and for $m > N$

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