An RKHS enjoys the following desirable smoothness property: if (left(g_{n} ight)) is a sequence belonging to RKHS

An RKHS enjoys the following desirable smoothness property: if \(\left(g_{n}\right)\) is a sequence belonging to RKHS \(\mathscr{G}\) on \(\mathscr{X}\), and \(\left\|g_{n}-g\right\|_{\mathscr{G}} \rightarrow 0\), then \(g(\boldsymbol{x})=\lim _{n} g_{n}(\boldsymbol{x})\) for all \(\boldsymbol{x} \in \mathscr{X}\). Prove this, using Cauchy-Schwarz.