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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.

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up vote 5 down vote accepted

Hint 1:

Jensen's inequality says that $$ \left(\int_0^1|f_n(x)-f(x)|\;\mathrm{d}x\right)^2\le\int_0^1|f_n(x)-f(x)|^2\;\mathrm{d}x $$ Hint 2:

Consider $$ f_n(x)=\left\{\begin{array}{}n^{2/3}&\text{when }x\le\frac1n\\0&\text{when }x>\frac1n\end{array}\right. $$

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Thank you this is excellent. – Galois Feb 21 '12 at 3:50

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