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Let $l^p(\mathbb{N})=\left\{ \{x_n\}_{n=1}^{\infty} : \|x\|_p=\left(\sum\limits_{n=1}^{\infty}|x_n|^p\right)^{1/p} < \infty \right\}$ with $1 \leq p < \infty$.

I would like some insight on how to show that this is a Banach space. I know that in order to be a Banach space that it must be complete. So I would have to show every Cauchy sequences converges.

I have to admit that I need some insight as to how to even start this.

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    You do not need to show (and it's not the case) that every sequence is Cauchy. You need to show that every Cauchy sequence has a limit.2011-10-13
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    Are you sure...the question was to prove that it is indeed a Banach space.2011-10-13
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    @wrldt: I think Alon is saying that you have the definition of completeness wrong. Completeness means that every Cauchy sequence is a convergent sequence.2011-10-13
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    Oh ok...that's my fault. I wrote this really quickly.2011-10-13
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    There is a proof here: www-math.mit.edu/~katrin/teach/18.100/LpCompleteness.pdf2011-10-13
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    Your last sentence ("I have to admit...") suggests that it should be the first step to check the definition of a Cauchy sequence. Then write down a sequence, assume it is Cauchy and try to deduce as much as you can...2011-10-13

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