Applying Arzela-Ascoli to show pointwise convergence on $\mathbb{R}$.

By Arzela-ascoli, there's a subsequence $f_n^{(1)}$ that converges uniformly on $[-1, 1]$. Then there's a subsequence $f_n^{(2)}$ of $f_n^{(1)}$ which converges uniformly on $[-2, 2]$. Similarly we define $f_n^{(k)}$. Now the sequence $$f_1^{(1)}, f_2^{(2)}, f_3^{(3)}, \dots $$ clearly converges uniformly on every interval $[-m, m]$, so the first condition is satisfied.

For the second part, for every $x \in \mathbb{R}$, $x \in [-m, m]$ for some $m$. Thus the sequence converges at $x$ to some limit $L_x$. We'll define $f(x) = L_x$. It should be easy to see that $f$ is continuous everywhere.