Let $X$ be a locally convex space with the family of seminorms $\{p_\alpha\}$. I am trying to get a feel for convergence of nets (or sequences) in these spaces aside from the general topological definition of convergence.
An answer to an older question (Convergence on locally convex spaces) mentions that convergence in locally convex spaces can be characterized by: $x_\beta \rightarrow x$ in a locally convex space if and only if $\rho_\alpha(x_\beta, x) \rightarrow 0$ for every seminorm $\rho_\alpha$.
Does anyone know a reference where this result may be found or how it can be proved?
Assume $x_\beta \to x$. Let $\varepsilon>0$. Since $$U=\{y:\rho(y-x)<\varepsilon\}$$ is an open neighborhood of $x$, there is some $\beta_0$ such that for all $\beta\ge\beta_0,\;x_\beta\in U$. This means for all $\beta\ge\beta_0,\;\rho(x_\beta-x) < \varepsilon$. Since $\varepsilon>0$ is arbitrary, it follows that $\rho(x_\beta-x)\to 0$.
Conversely, assume $\rho(x_\beta-x)\to 0$ for all seminorms $\rho$ defining the topology. A basic open neighborhood of $x$ has the form $$V=\{y:\rho_1(y-x)<\delta,\dots,\rho_n(y-x)<\delta\}$$ for some seminorms $\rho_1,\dots\rho_n$ and $\delta>0$. (Note: the actual definition of the topology uses different $\delta_i$ for $\rho_i$, but you can take $\delta=\min_i \delta_i$.)
Since $\rho_k(x_\beta-x)\to 0$ for each $k$, there is a $\beta_k$ such that $\rho_k(x_\beta-x)<\delta$ for $\beta\ge\beta_k$. By the property of nets, there is some $\gamma\ge\beta_k$ for all $k$. Then for all $\beta\ge\gamma$, for all $k$, $\rho_k(x_\beta-x)<\delta$. So for all $\beta\ge\gamma,\,x_\beta\in V$. Since $V$ was arbitrary, $x_\beta\to x$.