I have problems understanding parts of a proof in [Proposition 18.3.5., Diximier, C*-algebras] for the special case of discrete groups.
Let $G$ be a discrete group and let $\phi\in \ell^2(G)$. By 13.8.6. there exists a convolution square root, i.e. a function $\psi \in \ell^2(G) $ such that $\phi = \psi*\tilde{\psi}$ where * denotes the convolution product and $$\tilde{\psi} (s) =\overline{\psi(s^{-1} )}.$$ Let $(f_n) _n\subset c_c(G) $ be compactly supported functions that converge to $\psi$ in $\ell^2(G)$.
THEN $f_n *\tilde{f_n} $ converges uniformly to $\phi = \psi*\tilde{\psi}$. This is the part I can't understand.
Thank you in advance for any explanations and pointers.
For any $s\in G$, we have \begin{align*} |f_n*\tilde{f_n}(s)-\psi*\tilde{\psi}(s)|&\leq\sum_{t\in G}\left|f_n(t)\overline{f_n(s^{-1}t)}-\psi(t)\overline{\psi(s^{-1}t)}\right|\\ &\leq\sum_{t\in G}|f_n(t)||f(s^{-1}t)-\psi(s^{-1}t)|+\sum_{t\in G}|\psi(s^{-1}t)||f_n(t)-\psi(t)|\\ &\leq\|f_n\|_2\|f_n-\psi\|_2+\|\psi\|_2\|f_n-\psi\|_2. \end{align*} Let $\varepsilon>0$ be given. Since $f_n\to\psi$ in $\ell^2(G)$, there exists $N\in\mathbb N$ (not depending on $s$) such that $\|f_n-\psi\|_2<\min\{1,\frac{\varepsilon}{2\|\psi\|_2+1}\}$ for $n\geq N$. For such $n$, we have \begin{align*} |f_n*\tilde{f_n}(s)-\psi*\tilde{\psi}(s)|<(\|\psi\|_2+1)\|f_n-\psi\|_2+\|\psi\|_2\|f_n-\psi\|_2<\varepsilon \end{align*} and the result follows.