Sum of Cauchy Sequences in Topological Vector Space is Cauchy Sequence

Theorem
Let $K$ be a topological field.

Let $X$ be a topological vector space over $K$.

Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be Cauchy sequences in $X$.

Then $\sequence {x_n + y_n}_{n \in \N}$ is a Cauchy sequence in $X$.

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

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods: Corollary, there exists an open neighborhood $V$ of ${\mathbf 0}_X$ such that:
 * $V + V \subseteq U$

Since $\sequence {x_n}_{n \in \N}$ is Cauchy, there exists $N_1 \in N$ such that:
 * $x_n - x_m \in V$ for $n, m \ge N_1$.

Since $\sequence {y_n}_{n \in \N}$ is Cauchy, there exists $N_2 \in \N$ such that:
 * $y_n - y_m \in V$ for $n, m \ge N_2$.

Let $N = \max \set {N_1, N_2}$.

Then, for $n, m \ge N$, we have:

Hence $\sequence {x_n + y_n}_{n \in \N}$ is a Cauchy sequence in $X$.