Multiple of Vector in Topological Vector Space Converges

Theorem
Let $K$ be a topological field.

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

Let $x \in X$.

Let $\lambda \in K$.

Let $\sequence {\lambda_n}_{n \in \N}$ be a sequence in $K$ such that $\lambda_n \to \lambda$.

Then:


 * $\lambda_n x \to \lambda x$

Proof
From the definition of a topological vector space:


 * $\struct {\lambda, y} \mapsto \lambda y$

is continuous.

From Horizontal Section of Continuous Function is Continuous, we therefore have:


 * $\lambda \mapsto \lambda x$

is continuous.

From Continuous Mapping is Sequentially Continuous, $\lambda \mapsto \lambda x$ is sequentially continuous.

So we have:


 * $\lambda_n x \to \lambda x$