Give an example of series $f_n \in C[0,1]$ such that $f_n$ is Cauchy sequence in norm $$\|(a_n)\|_p = \left( \sum_{n=1}^{\infty} |a_n|^p \right)^{1/p}$$ and $$\lim_{n \to \infty} f_n(x)$$ don't exsists in $C[0,1]$.
For me the very big problem is to find Cauchy sequence in this norm. Of course, I know definition of Cauchy sequence. Could you give me an example of series which should I consider?
You should be able to find examples in nearly every text books. The standard example is some kind of ramp function, for instance this continuous approximation of $sign(x-\tfrac12)$ $$ f_ε(x)=\begin{cases} -1& 0\le x<\tfrac12-ε\\ \tfrac1ε(x-\tfrac12)&\tfrac12-ε\le x\le \tfrac12+ε\\ 1&\tfrac12+ε<x\le 1 \end{cases} $$