Continuity of sum of functions Suppose we have that $f_n:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and $f_n\geq0$ for all $n\in\mathbb{N}$. Assume that $f_n(x)\leq1$ for all $x\in[-n,n]$. My question is: Why is the function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined by $f=\sum_{n\in\mathbb{N}}{\frac{f_n}{2^n}}$ continuous? I thought about to use the Weierstrass M-test, but you do not know that all functions are bounded. Can someone help me with this problem?

Appy M-test the the tail of the series $\sum_{k=n}^\infty{f_k\over 2^k}$ in $[-n,n]$. $f =$ finite sum of continuous functions in $\Bbb R$ + continuous tail in $[-n,n]$ is continuous in $[-n,n]$. So, $f$ continuous in $\Bbb R$.
