Convergence on locally convex spaces

First you need to define the topology $\mathscr{T}$ on $X$. For each seminorm $\rho\in S$, $\epsilon>0$, and point $x\in X$, define the open ball $$ B(x,\rho,\epsilon)=\{y\in X\ |\ \rho(y-x)<\epsilon\}\ . $$ Take the collection of all such balls. That is a subbasis for the topology $\mathscr{T}$. Namely, by taking finite intersections and then arbitrary unions of these balls you get all the open subsets of $X$.

Convergence is defined in the usual way. A sequence $(x_n)$ in $X$ converges to some $x\in X$, iff $$ \forall V\ {\rm open\ neighborhood\ of\ } x, \exists N, \forall n\ge N,\ x_n\in V $$ Similarly unsurprising is the definition of $(x_n)$ being Cauchy: $$ \forall V\ {\rm open\ neighborhood\ of\ } 0, \exists N, \forall m,n\ge N,\ x_m-x_n\in V $$

That being said, a few remarks are in order. In this general setting, sequences are not the best thing to work with, especially if $S$ is uncountable. You should use nets instead, with similar definitions for being Cauchy and convergence. A good introductory reference is the book by Osborne on "Locally Convex Spaces". To go a bit further, "Introduction to Functional Analysis" by Meise and Vogt is also good.

A basis for the topology on $(X,S)$ consists of the sets of the form $$\{x \in X : \|x-x_0\|_1 < \varepsilon, \ldots, \|x-x_0\|_n < \varepsilon\}$$ for some $x_0 \in X$, $\varepsilon > 0$ and a finite collection of seminorms $\|\cdot\|_1, \ldots, \|\cdot\|_n \in S$.

Using this you can show that a net $(x_\lambda)_{\lambda\in \Lambda}$ converges to $x \in X$ if and only if $\|x- x_\lambda\| \to 0$ for every seminorm $\|\cdot\| \in S$.

Similarly, a net $(x_\lambda)_{\lambda\in \Lambda}$ is Cauchy if and only if for every $\varepsilon > 0$ and seminorm $\|\cdot\| \in S$ there exists $\lambda_0 \in \Lambda$ such that $\lambda, \mu \ge \lambda_0 \implies \|x_\lambda-x_\mu\| < \varepsilon$.