Proving the Continuity of $e^x$

Infinitely many continuous functions added together doesn't imply the result is also continuous. Take the Fourier series of a square wave function for example. All terms are continuous but the result is not. $$f(x)=\frac{4}{\pi}\sum_{n=1,3,5,...}{\frac{1}{n}\sin(\frac{n\pi x}{L})}$$

Here's another approach. Since $\exp x:=\sum_{n\ge0}\frac{x^n}{n!}$ converges for all $x$ by the ratio test, and you can show $\exp x\exp h=\exp (x+h)$ with the binomial theorem, we need only show continuity at $0$, since if $|h|<\delta\implies|\exp h-1|<\frac{\epsilon}{\exp x}$ then $|h|<\delta\implies|\exp(x+h)-\exp x|<\epsilon$. By the triangle inequality,$$|\exp h-1|=\left|\sum_{n\ge1}\frac{h^n}{n!}\right|\le\sum_{n\ge1}\frac{|h|^n}{n!}\le2\sum_{n\ge1}\left(\frac{|h|}{2}\right)^n=\frac{2|h|}{2-|h|}\le 2|h|$$for small $h$.