Linear Combination 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 $x, y \in X$ and $\lambda, \mu \in K$.

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

$\lambda x_n + \mu y_n \to \lambda x + \mu y$

Proof

From Scalar Multiple of Convergent Sequence in Topological Vector Space is Convergent, we have that:

$\sequence {\lambda x_n}_{n \in \N}$ converges to $\lambda x$

and:

$\sequence {\mu y_n}_{n \in \N}$ converges to $\mu y$.

From Sum of Convergent Sequences in Topological Vector Space is Convergent, $\sequence {\lambda x_n + \mu y_n}_{n \in \N}$ converges to $\lambda x + \mu y$.

$\blacksquare$