Here is a Lemma in a book "C*-algebras and Finite-Dimensional Approximations" P238.
Definition 7.1.1 A C*-algebra $A$ is called quasidiagonal (QD) if there exists a net of c.c.p. maps $\phi_{n}: A\rightarrow M_{k_{(n)}}(\mathbb{C})$ which is both asymptotically multiplicative (i.e., $||\phi_{n}(ab)-\phi_{n}(a)\phi_{n}(b)||\rightarrow 0$ for all $a, b\in A$) and asymptotically isometric (i.e., $||a||=limi_{n\rightarrow\infty}||\phi_{n}(a)||$ for all $a\in A$).
Lemma 7.1.4. If $A$ is unital and QD, then there exist u.c.p maps $\phi_{n}:A \rightarrow M_{k(n)}(\mathbb{C})$ which are both asymptotically multiplicative and asymptotically isometric.
Proof. Let $\phi_{n}^{'}: A\rightarrow M_{l(n)}(\mathbb{C})$ be asymptotically multiplicative and isometric c.c.p. maps. Functional calculus shows that the spectra of the matrices $\phi_{n}^{'}(1_{A})$ are contained in sets of the form $[0, \varepsilon_{n})\cup(\varepsilon_{n},1]$, where $\varepsilon_{n}\rightarrow 0$, and hence $$||\phi_{n}^{'}(1_{A})-P_{n}||\rightarrow 0,$$ where $P_{n}\in M_{l(n)}(\mathbb{C})$ are the spectral projections of $\phi_{n}^{'}(1_{A})$ corresponding to $[1, 1/2]$.
Thus $\phi_{n}^{'}(1_{A})P_{n}$ is an invertible element in $P_{n}M_{l(n)}(\mathbb{C})P_{n}$ and more functional calculus shows $$||(\phi_{n}^{'}(1_{A})P_{n})^{-\frac{1}{2}}-P_{n}||\rightarrow 0$$ as well. If $k(n)$ is the rank of $P_{n}$, then we get the desired u.c.p maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ by defining $$\phi_{n}(a)=(\phi_{n}^{'}(1_{A})P_{n})^{-\frac{1}{2}}\phi_{n}^{'}(a)(\phi_{n}^{'}(1_{A})P_{n})^{-\frac{1}{2}}.$$
My question:
In the second line of the proof, how does the author use the functional calculus shows that the spectra of the $\phi_{n}^{'}(1_{A})$ are contained in sets of the form $[0, \varepsilon_{n})\cup(\varepsilon_{n},1]$, where $\varepsilon_{n}\rightarrow 0$, and then how to get $$||\phi_{n}^{'}(1_{A})-P_{n}||\rightarrow 0 ?$$
From $\|\phi'_n(1_A)^2-\phi'_n(1_A)\,\|<\varepsilon_n$ we deduce that all spectral elements of $\phi'_n(1_A)$ satisfy the equation $|\lambda^2-\lambda|<\varepsilon_n$. As we are talking positive operators here, we get that $\sigma(\phi'_n(1_A))\subset [0,g(\varepsilon_n))\cup(1-g(\varepsilon_n),1]$, where $g$ is a function with $g\geq0$ and $\lim_{t\to0}g(t)=0$; properly, we can take $$ g(t)=\frac{2t}{1+\sqrt{1-4t}}. $$ There is no part above $1$ because $\phi'_n$ is completely contractive.
The inequality follows by functional calculus from the inequality $|t-1_{[1/2,1]}(t)|<g(\varepsilon_n)$ when $t\in [0,g(\varepsilon_n))\cup(1-g(\varepsilon_n),1]$: it implies that the selfadjoint operator $\phi'_n(1_A)-P_n$ has its spectrum inside $[0,g(\varepsilon_n))$.