Proposition 8

Let $E$ be a Hausdorff locally convex space and let $A$ be a closed, balanced, bounded and convex subset of $E$. Let $(x_n)$ be a Cauchy sequence in $E_A$. Then this sequence converges in $E_A$ if and only if it converges in $E$.

The canonical injection from $E_A$ into $E$ is continuous. Hence, if $(x_n)$ converges in $E_A$, it converges in $E$. Conversely, suppose $(x_n)$ converges to $y$ in $E$. There exists an increasing sequence of integers $(n_k)$ such that $p_A(x_m - x_n) \leq 2^{-k-1}$ if $m \geq n_k$ and $n \geq n_k$. Therefore the sequence $(x_{n_k} + 2^{-k}A)$ is decreasing. Since $A$ is closed in $E$, we have $y \in \bigcap_k (x_{n_k} + 2^{-k}A)$, which proves that $(x_{n_k})$, hence $(x_n)$, converges to $y$ in $E_A$.