Here is a lemma about quasicentral-approximate-unit:
Lemma 7.3.1Let $J\triangleleft A$ be a separable ideal. Then there exists a quasi-central approximate unit $\{e_{j}\}\subset J$ such that $e_{j+1}e_{j}=e_{j}$ for all $j\in \mathbb{N}$.
Proof. Since $J$ is separable, it contains a strictly positive element $h$ (i.e., $\phi(h)>0$ for all states $\phi$ on $J$-let $h=\sum\frac{1}{2^{j}}e_{j}$ where $\{e_{j}\}$ is any approximate unit). For each $n$ let $f_{n}\in C_{0}(0, 1]$ be the function which is zero on the intercal (0, $\frac{1}{2^{n}}]$, one on the intercal [$\frac{1}{2^{n-1}}$, 1] and linear in between. Evidently we have $f_{n+1}(h)f_{n}(h)=f_{n}(h)$, for all $n$, and it is also readily seen that $\phi(f_{n}(h))\rightarrow 1$ for all states $\phi$ on $J$. The usual Habh-Banach convexity argument allows us to extract a quasicentral approximate unit from the convex hull of $\{f_{n}(h)\}$, so we leave the remaining details to you.
My question is
In the end of the proof, how to usethe Hahn-Banach convexity argument to find a quasicentral approximate unit?
You have to look at Theorem 1.2.1, which is just a re-statement of I.9.16 in Davidson's book. Davidson's proof is a page long, not counting auxiliary results.
Davidson's proof doesn't state it explicitly, but the condition $e_{j+1}e_j=e_j$ can be achieved by going far enough along the index set when defining $\mathcal F$ in Davidson's proof.