A very slowly convergent sequence of continuous functions

Abstract:

A sequence of continuous functions ${f_n}:[0,1] \to (0,1]$ is constructed, with ${\lim _{n \to \infty }}{f_n}(x) = 0$ for every $x \in [0,1]$, but such that to every unbounded sequence $\{ {\lambda _n}\}$ of positive numbers corresponds a point $x \in [0,1]$ at which $\lim {\sup _{n \to \infty }}{\lambda _n}{f_n}(x) = \infty$.