I have the following theorem about the criterion of compactness in $l_p$ space
For the set $K\subset (l_p,||.||_p), p\geq 1$, following conditions are equivalent:
i) $K$-totally bounded in $(l_p,||.||_p)$;
ii) $K$ is bounded in respect to the norm $||.||_p$ and for any $\epsilon>0$ there exists a number $n(\epsilon)$, s.t $\sum\limits_{i=n(\epsilon)+1}^{\infty}|x_i|^p<\epsilon^p$, for any $x=\{x_n\}\in K.$
Here is the proof for $ii)\Rightarrow i)$
I don't understand why the book had to prove $||x-a_n(x)||_p<\epsilon$? And what is the point of proving of the set $K_n$ is $\epsilon$-net in the space $l_p$?
(I understand that we can't conclude $K$ is totally bounded if the set $K_n$ is $\epsilon$-net in the space $l_p$ since we don't know $K_n$ is finite or not?)