Let $f$ and $f_n$ for $n=1,2,\ldots,n$ be Riemann integrable real-valued functions defined on $[0,1]$. For each of the following statements, determine whether the statement is true or not and prove you claim: (a) If $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|dx=0$ then $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|^2dx=0$. (b) If $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|^2dx=0$, then $\lim_{n\to\infty}\int_{0}^{1}\left|f_{n}(x)-f(x)\right|dx=0$.
I have no idea how to approach this problem. I tried to think of counter examples but didn't succeed. Any help would be much appreciated.
This is not a homework problem, it is a practice problem for a midterm. Any help would be much appreciated.