Failure of convergence to 0

Question

Consider the interval $I = [0,1]$ and the sequence of functions:

$$f_n(x) = (-1)^k \ \text{for} \displaystyle \frac{k}{2^n} \le x < \frac{k+1}{2^n} \ \text{where} \ 0 \le k < 2^n - 1$$

I want to exhibit that $f_n \not \to f \equiv 0$ strongly in $L^p[I]$ $\forall 1 \le p < \infty$

Answer 1

The definition of the $L_p$ norm of $f_n$ is $\|f_n\|_p=\left(\int_I |f_n(x)|^pdx\right)^{1/p}$. If $y=1$ or $y=-1$, then $|y|^p=1$. What is $\|f_n\|_p$?

The definition of convergence of a sequence $(f_n)$ to zero in $L^p(I)$ says that $\lim\limits_{n\to\infty}\|f_n\|_p=0$. Does this happen for $(f_n)$ in your question?

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The sequence does converge weakly to $0$.