Prove that this sequece converges in $L^1$, but does not converges a.e. in $\mathbb{R}$

Question

Well, considering the sequence: $$ f_n(x) = \chi_{[j2^{-k},(j+1)2^{-k}]}(x) $$ Where $\chi_A$ denotes the characterisct function of $A$, $n=j+ 2^k$ and $0\le j < 2^k$.

I showed that $f_n \to 0$ in $L^1(\mathbb{R})$, but I couldn't show that $f_n$ does not converge to $0$ a.e in $\mathbb{R}$.

Can anybody help me?

Answer 1

Hint: For any $a\in[0,1]$, the value $f_n(a)$ is $1$ for infinite many $n$'s.

(For every $k$, the intervals $[j/2^k,\, (j+1)/2^k]$ cover $[0,1]$, so $a$ must be in one of them.)