Let $\{f_n\}$ be a sequence of real valued continuous function on $(0,\infty)$ such that $f_n(x) = f_n(2x)$ for all $x$ for all $n$. Suppose $\{f_n\}$ is uniformly bounded can we conclude that $\{f_n\}$ has a uniformly converging subsequence?
Any such function $f_n$ is defined by its value in the interval $[1,2]$. I have tried to construct an example contradicting this statement. But I could not process further. Kindly give some hints.
Thank you.