Consider for $x \in \mathbb{R}$ the sum function defined as $$ f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1) $$ I have shown that the series converges point wise by using that $$ |exp(-x^2/n)| \leq |-x^2/n| = x^2/n $$ from an earlier question. The problem is now that to show that $f$ is continous on $\mathbb{R}$ from a sentence in my book I would have to show that $f$ has a convergent majorant series satisfying that $$ |f_n(x)| \leq M_n $$ for all $x \in \mathbb{R}$ but as I know that this is not possible unless $x$ is in a compact set. I thought about letting $x \in [-K,K]$ but I don't think I am allowed as I have to show that $f$ is continous for all $x \in \mathbb{R}$ which $x \in [-K,K]$ doesn't satisfy I suppose.
What can I do?
Showing that $f$ is continuous on $[-K, K]$ for all $K>0$ is enough, because for a given $x \in \mathbb{R}$, you can let $K=2|x|$, and have that $f$ is continuous on $[-2|x|, 2|x|]$, hence it's continuous at $x \in [-2|x|, 2|x|]$.