Cauchy Sequence is Bounded

Theorem
Every Cauchy sequence is bounded.

Proof
Let $$\left \langle {x_n} \right \rangle$$ be a Cauchy sequence.

That is, $$\forall \epsilon > 0: \exists N: \forall m, n > N: \left|{x_n - x_m}\right| < \epsilon$$.

Particularly, setting $$\epsilon = 1$$, we have $$\exists N_1: \forall m, n > N_1: \left|{x_n - x_m}\right| < 1$$.

Now let $$m = N_1 + 1$$.

By the Triangle Inequality, for any $$n > N_1$$:

$$ $$ $$

Now we take $$K = \max \left\{{\left|{x_1}\right|, \left|{x_2}\right|, \ldots, \left|{x_{N_1}}\right|, \left|{x_{N_1 + 1}}\right|}\right\}$$.

It follows that $$\forall n \in \mathbb{N}^*: \left|{x_n}\right| \le K$$ and so $$\left \langle {x_n} \right \rangle$$ is bounded.