Cauchy Sequence is Bounded/Normed Vector Space

Theorem
Let $V = \struct {X, \norm {\,\cdot\,} }$ be a normed vector space.

Every Cauchy sequence in $X$ is bounded.

Proof
Let $\sequence {x_n} $ be a Cauchy sequence in $V$.

Then by definition:
 * $\forall \epsilon \in \R_{>0}: \exists N \in \N : \forall n, m \ge N: \norm {x_n - x_m} < \epsilon$

Let $N$ satisfy:
 * $\forall n, m \ge N: \norm {x_n - x_m} < 1$

Let $m = N + 1 > N$.

Then $\forall n \ge N$:

Let $M = \max \set {\norm {x_1}, \norm {x_2}, \dots, \norm {x_N}, 1 + \norm {x_{N + 1} } }$.

Then:
 * $\forall n < N: \norm {x_n} \le M$
 * $\forall n \ge N: \norm {x_n} \le 1 + \norm {x_{N + 1} } \le M$

It follows by definition that $\sequence {x_n}$ is bounded.