( \{ \mathfrak{A}(n) \} )$ lifts to a projection in $\ell^{\infty}( \{ \mathfrak{A}(n) \} )$. By Lemma \[l:reducedBusby\], we have that $\mathfrak{A} \cong \mathfrak{A}( \infty ) \oplus _{ \overline{\tau} , \rho } \ell^{ \infty } ( \{ \mathfrak{A}(n) \} )$. It is now clear that every projection in $\mathfrak{A} (\infty)$ lifts to a projection in $\mathfrak{A}$. By Lemma \[l:liftingfdquot\], there exists a sequence of finite dimensional sub-$C^{*}$-algebras of $\mathfrak A$, $\{ \mathfrak B_k\}_{k=1}^\infty$, such that $\mathfrak B_k \cap \mathfrak A(\mathbb N)=0$ and $\bigcup_{ k  =1}^{\infty} ( \mathfrak{B}_{k} + \mathfrak{A}({\ensuremath{\mathbb{N}}\xspace}) )$ is dense in $\mathfrak{A}$. By the epimorphisms onto the direct summands $\pi_n\colon \mathfrak A \to \mathfrak A(n)$ for $n\in \mathbb N$, we get a $*$-homomorphism $\overline \sigma \colon \mathfrak A \to \ell^\infty( \{ \mathfrak A(n)\})$. Let $\{ p_m^n \}_{m=1}^{\infty}$ be an increasing approximate identity of projections in $\mathfrak A(n)$ for each $n\in \mathbb N$. By passing to subsequences we may assume that $$\| p_m^n x - xp_m^n \| < 1/m \text{ for } x\in \pi_n \left(\bigcup_{k=1}^m(\mathfrak B_k)_1\right), n\leq m,$$ since the closed unit balls $(\mathfrak B_k)_1$ are compact. Define $p_m := \sum_{n=1}^m p_m^n$. Clearly $\{ p_m\}_{m=1}^\infty$ is an approximate identity of projections in $\mathfrak A(\mathbb N)$. We claim that this 