Construct an example of a sequence of functions $(f_n)$ defined on $[0,1]$ such that $f_n$ converges pointwise to $0$ and for every sequence of numbers $(a_n)$ that tends to $\infty$, sequence $(a_nf_n)$ doesn't converge to $0$. Does anyone have any ideas? I am having trouble overcoming the fact we do not know the speed of $(a_n)$. What ever I come up with for $f_n$ I think I could find fast/slow enough $a_n$ to make $(a_nf_n)$ go to $0$. Obviously, this is not true. So, I would be very grateful for some help.
I found out this is an exercise in Rudin's Functional analysis. Chapter 1, exercise 7.