projection in $\mathfrak{A}$.

Suppose $\mathfrak{A}$ is a $C^{*}$-algebra with an ideal $\mathfrak{I}$ such that $p \mathfrak{I} p$ has an approximate identity consisting of projections for all projections $p \in \mathfrak{A}$, $\mathfrak{A} / \mathfrak{I}$ is an AF-algebra, every projection in $\mathfrak{A}/\mathfrak{I}$ lifts to a projection in $\mathfrak{A}$. Since $\mathfrak{A} / \mathfrak{I}$ is an AF-algebra, there exists an increasing sequence of finite dimensional sub-$C^{*}$-algebras $\{ \mathfrak{D}_{k} \}_{ k = 1}^{\infty}$ of $\mathfrak{A} / \mathfrak{I}$ such that $\mathfrak{A} / \mathfrak{I} = \overline{ \bigcup_{ k = 1}^{\infty } \mathfrak{D}_{k} }$. By the first part of the lemma, we have a sequence of $*$-homomorphisms, $\{ {  \phi_{k}  \colon  \mathfrak{D}_{k}  \rightarrow  \mathfrak{A}  }  \}_{ k = 1}^{\infty}$ such that $\pi \circ \phi_{k} = {\ensuremath{\operatorname{id}}}_{ \mathfrak{D}_{k}}$.

Set $\mathfrak{C}_{k} = \phi_{k}( \mathfrak{D} _{k} )$. Then $\mathfrak{C}_{k}$ is a finite dimensional sub-$C^{*}$-algebra of $\mathfrak{A}$. Since $\pi \circ \phi_{k} = {\ensuremath{\operatorname{id}}}_{ \mathfrak{D}_{k}}$, we have that $\mathfrak{C}_{k} \cap \mathfrak{I} = 0$. Note that $\mathfrak{C}_{k} + \mathfrak{I} \subseteq \mathfrak{C}_{k+1} + \mathfrak{I}$ since $\mathfrak{D}_{k} \subseteq \mathfrak{D}_{k+1}$. Let $x \in \mathfrak{A}$ and let $\epsilon > 0$. Since $\pi ( \bigcup_{ k =1}^{\infty} \mathfrak{C}_{k} ) = \bigcup_{ k = 1}^{\infty} \mathfrak{D}_{k}$, there exists $y_{1} \in \mathfrak{C}_{k}$ (for some $k$) such that $\| \pi(x) - \pi( 