Linear Combination 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 $X$.

Let $\lambda, \mu \in \GF$.

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

Proof

From Scalar Multiple of Cauchy Sequence in Topological Vector Space is Cauchy Sequence, $\sequence {\lambda x_n}_{n \in \N}$ and $\sequence {\mu y_n}_{n \in \N}$ are Cauchy in $X$.

From Sum of Cauchy Sequences in Topological Vector Space is Cauchy Sequence, $\sequence {\lambda x_n + \mu y_n}_{n \in \N}$ is Cauchy.

$\blacksquare$