Sum of Cauchy Sequences in Topological Vector Space is Cauchy Sequence

Theorem

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 $X$.

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

$V + V \subseteq U$

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:

$\ds \paren {x_n + y_n} - \paren {x_m + y_m}$

$=$

$\ds \paren {x_n - x_m} + \paren {y_n - y_m}$

$\ds $

$\in$

$\ds V + V$

$\ds $

$\subseteq$

$\ds U$

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

$\blacksquare$