Cauchy Sequence is Bounded/Real Numbers/Proof 1

Theorem

Every Cauchy sequence in $\R$ is bounded.

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 $m, n \ge N$.

So for all $m \ge N$, we have:

$\ds \size {a_m}$

$=$

$\ds \size {a_N + a_m - a_N}$

$\ds $

$\le$

$\ds \size {a_N} + \size {a_m - a_N}$

$\ds $

$\le$

$\ds \size {a_N} + 1$

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

$\blacksquare$