$l^{p}(\mathbb{N})$ is a Banach Space 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.
 A: Given a Cauchy sequence $(x^n)_{n\in\mathbb N}$ in $\ell^p(\mathbb N)$, it is a good idea to check that for each $m\in\mathbb N$ the sequence in the $m$-th coordinate, $(x_m^n)_{n\in\mathbb N}$ is Cauchy and hence, since $\mathbb C$ is complete, converges.  Now guess what the limit of the sequence $(x^n)_{n\in\mathbb N}$ might be.
A: Proving this space is complete requires you to show that every Cauchy sequence converges in the space. Why don't Cauchy sequence always converge in the space? Consider the set $(0,1) \subset \mathbb{R}$ and think of a Cauchy sequence that doesn't converge in the set $(0,1)$.
So start with a Cauchy sequence, say, $\{x_n\}_{n=1}^{\infty}$. What are each of the $x_i$'s? Given an $\epsilon > 0$ what can you say about this sequence?  
Since completion of $\mathcal{\ell}^p(\mathbb{N})$ means $\{x_n\}$ converges, you had better have a good guess as to what it converges to. You have to "guess" what it converges to.
You know some things about what $\{x_n\}$ should converge to: you know it should be in $\mathcal{\ell}^p(\mathbb{N})$ (else the space wouldn't be complete and you'd be trying to prove a false statement) and you know that it must involve limits and the sequence $\{x_n\}$ somehow. Call this object $y$.
My hint on how to pick $y$ is to remember that it is itself a sequence (why is this the case?)! Think about $y$ coordinate by coordinate and think about what "should happen" if $x_n \rightarrow y$. If you pick your $y$ correctly, it should be in the space $\mathcal{\ell}^p$ "for free" (or "by construction").
Once you have this sequence you aren't done; you need to show it is a limit in your space. Whatever you did in the previous step possibly involved limits in $\mathbb{R}$, but you need to show your sequence $\{x_n\}$ converges to $y$ in the space $\mathcal{\ell}^p$. 
What does it mean to converge in $\mathcal{\ell}^p$? How is it different than convergence in $\mathbb{R}$?
Once you figure out the answers to those questions, you should proceed to show that, indeed, $\{x_n\} \rightarrow y$ in $\mathcal{\ell}^p$.
