Suppose that $f\in L^1(\mathbb{R})$ and that $f(x)=0$ if $x\notin[0,1]$. For $n=1,2,\ldots,$ let $f_n(x)=f\left(x+\frac{1}{n}\right)$. Prove that $\|f_n-f\|_1\to 0$ as $n\to\infty$.
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This is true for any $f \in L^1$, even without the additional assumption. See here: http://math.stackexchange.com/questions/157397/proof-that-translation-of-a-function-converges-to-function-in-l1 – PhoemueX Mar 08 '15 at 09:08