I wanna to prove the completeness of Schwartz space $\mathscr{S}(\mathbb{R}^{n})$ equipped with the induced topology from a set of seminnorms $$\|f(x)\|_{\alpha,\beta}=\sup_{x\in \mathbb{R}^{n}}|x^{\beta}D^{\alpha}f(x)|$$
A Cauchy sequence in Schwartz space is a sequence of $\{f_{i}\}$ such that $$\lim_{k\to\infty}\|f_{i}(x)-f_{j}(x)\|_{\alpha,\beta}=0 \quad\quad \forall i,j\gt k, \alpha,\beta \in \mathbb{Z}_{+}^{n}$$
Since the seminorms are uniform norm, for each $\alpha,\beta$, there must be a pointwise limit function $f_{\alpha,\beta}$ such that
$$x^{\beta}D^{\alpha}f_{i}(x)\to f_{\alpha,\beta}(x) \quad as \quad i\to\infty$$
My question is how I can show that there must be a function $f(x)\in\mathscr{S}(\mathbb{R}^{n})$ such that for each $\alpha,\beta$, $x^{\beta}D^{\alpha}f(x)=f_{\alpha,\beta}(x)$?
Note that your convergence is uniform. For a sequence of differentiable functions $\{f_n\}$, if $\{f'_n\}$ converges uniformly, and $\{f_n(x)\}$ converges for some $x$, then $f_n\to f$ and $f_n'\to f'$.