Rademacher function and weak convergence

Question

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.

a) Show that $r_{n}\xrightarrow{w}0$ in $L_1$;

b) Does $r_{n}\xrightarrow{w}0$ in $L_p$ for $1<p<\infty$?

c) Show that in $L_{\infty}$, the sequence $(r_n)$ spans an isometric copy of $\ell_1$.

I'm really stuck and I don't know how to begin, any ideas or hints please. Thank you.