\alpha_g (a)\right\|
+
\left\| \alpha_{g} (a) - z_{n}(g) a z_{n}(g)^*\right\|
\\
& \qquad+
\|z_{n}(g)\| \cdot \|  a  -  x \| \cdot \|z_{n}(g)^*\|
\\
&<
\frac{{\varepsilon}}{3} + \frac{{\varepsilon}}{3} + \frac{{\varepsilon}}{3} ={\varepsilon}.\end{aligned}$$ Thus, $$\lim_{n \to \infty}
 \left\| \alpha_{g} (x) - z_{n}(g) x z_{n}(g)^*\right\|=0.$$ Therefore, for all $x \in A$, and all $g \in G$, $$\alpha_{\infty g} (x) = w(g) x w(g)^*.$$ By (\[D\_9719\_AppRep.2’\]) and (\[D\_9719\_AppRep.3’\]), we further have, for all $g, h \in G$, $$\lim_{n \to \infty}
 \left\| z_{n}(g) z_{n}(h) - z_{n}(gh)\right\|=0
\quad 
\mbox{ and }
\quad 
\lim_{n \to \infty}
 \left\| \alpha_{g} (z_n (h)) - z_{n}(g h g^{-1})\right\|=0.$$ Thus, the map $g \mapsto w(g)$ is a unitary representation from $G$ to $A_{\infty}$ and $\alpha_{\infty g} (w(h)) = w(g h g^{-1})$ for all $g, h \in G$.

To prove the forwards implication, consider $w(g)$ in Definition \[Izum\_Def\_app\_Rep\] as being presented by a sequence $(w_n (g))_{n \in {{\mathbb{Z}}_{> 0}}}$. Then $$\lim_{n \to \infty} \|w_n(g) w_n (g)^* -1 \|=0
\qquad
\mbox{ and }
\qquad
\lim_{n \to \infty} \|w_n (g)^* w_n(g) -1 \|=0.$$ Now using semiprojectivity to $C(S^1)$, we can find a sequence $(u_n (g))_{n \in {{\mathbb{Z}}_{> 0}}}$ of unitaries in $A$ such that $\lim_{n \to \infty} \|u_n(g) - w_n(g)\|=0$. Then we still have, for all 