for any finite self-adjoint subset $F\subset A_m$ and $\varepsilon >0$, there exists $n\geq m$ such that $$\phi_{m,n}(F)\subset_{\epsilon}\{ f\in (A_n)_{sa}\,|\,f\,has\,finite\,spectrum\}.$$

The following fact is well known. (called Pairing Lemma[@Su] and Marriage Lemma in [@HV])

Let $X,Y$ be two sets of finitely many points in a metric space, $\#(X)=\#(Y)$. Suppose that for any subset $\widetilde{X}\subset X$, $$\#\big \{y\in Y,\, dist(y,\widetilde{X})<\varepsilon\big \}\geq\#(\widetilde{X}),$$ and any subset $\widetilde{Y}\subset Y$, $$\#\big\{x\in X,\, dist(x,\widetilde{Y})<\varepsilon \big \}\geq\#(\widetilde{Y}).$$ Then $X$ and $Y$ can be paired to within $\varepsilon$ one by one.

We need the following generalization of the pairing lemma due to Gong (see [@EGLN])

\[2.5\] Let $X,Y$ be two sets of a metric space with subsets $X_1\subset X$ and $Y_1\subset Y$. Suppose that for each subset $\widetilde{X}\subset X_1$, $$\#\big \{y\in Y,\, dist(y,\widetilde{X})<\varepsilon\big \}\geq\#(\widetilde{X})$$ and for each subset $\widetilde{Y}\subset Y_1$, $$\#\big\{x\in X,\, dist(x,\widetilde{Y})<\varepsilon \big \}\geq\#(\widetilde{Y}).$$ Then there are subsets $X_0\subset X$ ,$Y_0\subset Y$ with $X_0\supset X_1$, $Y_0\supset Y_1$ such that $X_0$ and $Y_0$ can be paired to within $\varepsilon$ one by one.

\[2.6\] Let $A\in \mathcal{C}$, for any $1>\varepsilon >0$ and $\eta=\frac{1}{m}$ where $m\in\mathbb{N}_+$, if $\phi,\psi:A\rightarrow M_n(\mathbb{C})$ are unital homomorphisms with the condition that $Eg(\phi(h))$ and $Eg(\varphi(h))$ can be paired to within $\frac{1}{4}$ one by one for all $h\in 