Cauchy Sequence is Bounded/Normed Vector Space

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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 $\forall n \ge N$:

\(\ds \norm {x_n}\) \(=\) \(\ds \norm {x_n - x_{N + 1} + x_{N + 1} }\)
\(\ds \) \(\le\) \(\ds \norm {x_n - x_{N + 1} } + \norm {x_{N + 1} }\) Norm Axiom $\text N 3$: Triangle Inequality
\(\ds \) \(\le\) \(\ds 1 + \norm {x_{N + 1} }\)


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.

$\blacksquare$


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