Possible Duplicate:
How to show that this set is compact in $\ell^2$
Here's my problem:
Let $p \geq 1$ and let $(r_k)_{k=1}^\infty$ be a sequence in $\mathbb{R}$ such that $r_k > 0$ for all $k \in \mathbb{N}$ and $\sum_{k=1}^\infty r_k < \infty$. Show that
$ K = \{\{x_k\} \in l_p: |x_k| \leq r_k \text{ for all } k \in \mathbb{N} \} $
is compact in $\ell^p$.
I want to do this by showing that $K$ is complete and totally bounded. The complete part is fairly simple since it's a closed subspace of a complete space. Now I'm doing the totally bounded part, so for any $\varepsilon$ we need to find a finite collection of $x \in \ell^p$ such that open $\varepsilon-$balls around these points cover $K$. For this I have:
Let $\varepsilon \in \mathbb{R}_{>0}$, and choose $N \in \mathbb{N}$ such that $\sum_{k=N}^{\infty}r_k < \frac{1}{2}\varepsilon$. Then $\sum_{k=N}^{\infty}|x_k| \leq \sum_{k=N}^{\infty}r_k < \frac{1}{2}\varepsilon$.
Now I'm stuck. We've got the tail covered, so now we have to worry about the the points coordinated $1$ through $N-1$, which are each bounded by $r_1$ through $r_{N-1}$. How do we then choose proper sequences to bound this?