Definition 2.3.1. A C*-algebra $A$ is nuclear if the identity map id$_{A}:A \rightarrow A$ is nuclear.
Exercise 2.3.7. If for each finite set $F\subset A$ and $\epsilon>0$ one can find a nuclear subalgebra $B\subset A$ such that $B$ almost contains $F$, within $\epsilon$ in norm, then $A$ is nuclear. In particular, the class of nuclear C*-algebras is closed under taking inductive limits with injective connecting maps.
My question are
1. I do not know the definition of "almost contain, within $\epsilon$ in norm".
2. Could someone give show me more details of this exercise or give me some hints?
Thanks