help with pointwise convergence and uniform convergence

Question

Consider D a finite set, and $f_{n\:}$ pointwise converge for every $d\in D$.
How to prove that $f_n$ converges uniformly in D?

Answer 1

Let $$D=\{d_1,\ldots,d_p\}$$ and $$f_n(d_k)\xrightarrow{n\to\infty}f(d_k),\quad k=1,\ldots,p$$ so $$\max_{1\le k\le p}\left|f_n(d_k)-f(d_k)\right|=\left|f_n(d_m)-f(d_m)\right|\xrightarrow{n\to\infty}0$$ and then the convergence is uniform.

Answer 2

By pointwise convergence, for each point in $d_i \in D$ and every $\epsilon > 0$, you get that there is some $N_i > 0$ so that $N > N_i$ implies that $|f_N(d_i) - f(d_i)| < \epsilon$. There are only finitely many $N_i$ so then you can take the max, call it $N^*$, and then you will have that $N > N^*$ implies that $|f_N(d_i) - f(d_i)| < \epsilon$ for all $d_i \in D$, so the convergence is uniform. Note this hinges crucially on the fact that $D$ is finite so that you can take the maximum over the $N_i$ without getting an infinite answer.