Rudin and Benyamini have proved the existence of sequences (the proofs, and the sequences, seem to be essentially different) of continuous, strictily positive and uniformly bounded sequence of functions $(f_{n})_{n\geq 1}$ defined on $[0,1]$ "very slowly convergent". Formally, such sequence satisfy the following conditions:
(1) $\lim_{n}f(x)=0$ for each $x\in[0,1]$ (or even $\sum_{n\geq 1}f_{n}(x)\leq 1$ for each $x\in [0,1]$.
(2) Given an unbounded sequence of positive numbers $(\lambda_{n})_{n\geq 1}$, there is $x\in [0,1]$ such that $\lim_{n}\lambda_{n}f_{n}(x)=+\infty$.
Really, such sequence of functions is very "nice". My question is the following: Somebody know some application for such sequence? That is, can we use these functions to show some result, say, on Topology or apply them to compute some "numerical" formula (series, integral, etc...)?
Many thanks for your time!