Characterization of Cauchy Sequence in Topological Vector Space in terms of Local Basis

Theorem

Let $\struct {X, \tau}$ be a topological vector space.

Let $\BB$ be a local basis for $\mathbf 0_X$ in $\struct {X, \tau}$.

Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence.

Then $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy if and only if:

for each $V \in \BB$ there exists $N \in \N$ such that $x_n - x_m \in V$ for $n, m \ge N$.

Suppose that $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy.

Then:

for each open neighborhood $V$ of $\mathbf 0_X$ there exists $N \in \N$ such that:

$x_n - x_m \in V$ for each $n, m \ge N$.

Since $\BB$ consists of open neighbourhoods of $\mathbf 0_X$, we in particular have:

for each $V \in \BB$ there exists $N \in \N$ such that $x_n - x_m \in V$ for $n, m \ge N$.

$\Box$

Suppose that:

for each $V \in \mathcal B$ there exists $N \in \N$ such that $x_n - x_m \in V$ for $n, m \ge N$.

Let $U$ be an open neighborhood of $\mathbf 0_X$.

Then there exists $V \in \mathcal B$ such that $V \subseteq U$.

Then there exists $N \in \N$ such that $x_n - x_m \in V \subseteq U$ for $n, m \ge N$.

So $\sequence {x_n}_{n \mathop \in \N}$ is Cauchy.

$\blacksquare$