Sum of Convergent Sequences in Topological Vector Space is Convergent

Theorem

Let $K$ be a topological field.

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

Let $x, y \in X$.

Let $\sequence {x_n}_{n \in \N}$ and $\sequence {y_n}_{n \in \N}$ be convergent sequences with:

$x_n \to x$

and:

$y_n \to y$

Then $\sequence {x_n + y_n}_{n \in \N}$ converges with:

$x_n + y_n \to x + y$

Proof

Let $W$ be an open neighborhood of $x + y$.

From Open Neighborhood of Point in Topological Vector Space contains Sum of Open Neighborhoods there exists an open neighborhood $U$ of $x$ and an open neighborhood $V$ of $y$ such that:

$U + V \subseteq W$

Since $\sequence {x_n}_{n \in \N}$ converges to $x$, there exists $N_1 \in \N$ such that:

$x_n \in U$ for $n \ge N_1$.

Since $\sequence {y_n}_{n \in \N}$ converges to $y$, there exists $N_2 \in \N$ such that:

$y_n \in V$ for $n \ge N_2$.

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

Then for $n \ge N$ we have:

$x_n + y_n \in U + V \subseteq W$

Since $W$ was an arbitrary open neighborhood of $x + y$, we have that $\sequence {x_n + y_n}$ converges to $x + y$.

$\blacksquare$