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Often when there is some inequality that we want to prove in a normed space, the proof goes something like "Assume there's a sequence $f_n$ with $\lVert f_n \rVert = 1$..."

Would someone give me a concrete example of this method of proof? I am not very familiar with it and need to study it.

Thanks

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Often, an inequality we want to prove is homogeneous (that is, if it works for $x$, it will be true for $\alpha x$ for any scalar $\alpha$. So when we argue by contradiction, we can assume that all the elements of the sequence we got have the same norm. –  Davide Giraudo Nov 19 '12 at 12:46
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