As reference the german wiki: Approximationseigenschaft
Problem
Given a Banach space.
Suppose it has the approximation property: $$C\in\mathcal{C}:\quad\|T_N-1\|_C\to0\quad(T_N\in\mathcal{F}(E))$$
Then every compact operator is of almost finite rank: $$\overline{\mathcal{F}(X,E)}=\mathcal{C}(X,E)\subseteq\mathcal{B}(X,E)$$
How do I prove this actual equivalence?
Attempt
As the image of the unit ball is precompact one has: $$C\in\mathcal{C}(X,E):\quad\|T_NC-C\|=\|T_N-1\|_{C(B)}\to0\quad(T_NC\in\mathcal{F}(X,E))$$ For the converse one might try to smuggle in a compact operator: $$C\subseteq rB:\quad\|T_N-1\|_C\leq r\|T_N-C\|_B+\|C-1\|_C<r\delta_T+\delta_C\quad(C\in\mathcal{C}(E))$$ But how to construct one that approximates the identity?
This is a nontrivial result by Grothendieck!
(See Lindenstrauss & Tzafriri, Theorem 1.e.4, Volume I.)