KöMaL problem A, 2017

Let $f_1,f_2,\ldots$ be an infinite sequence of continuous $\mathbb{R}\to\mathbb{R}$ functions such that for arbitrary positive integer $k$ and arbitrary real numbers $r>0$ and $c$ there exists a number $x\in(-r,r)$ with $f_k(x)\ne cx$. Show that there exists a sequence $a_1,a_2,\ldots$ of real numbers such that $\sum_{n=1}^\infty a_n$ is convergent, but $\sum_{n=1}^\infty f_k(a_n)$ is divergent for every positive integer $k$.