Let $G$ be a locally compact abelian, $T_2$ group such that $G,\hat{G}$ both are $\sigma$-compact with $G=\bigcup K_n$ and $\hat{G}=\bigcup C_n$ where $K_n,C_n$ are increasing compact subsets of $G$ and $\hat{G}$ respectively. Let $f\in L^2(G)$, define $$\phi_n(\chi)=\int\limits_{K_n} f(x)\chi(x^{-1})\ dx,\hspace{2mm} \psi_n(x)=\int\limits_{C_n}\hat{f}(\chi)\chi(x)\ d\chi$$ Show that, $\lVert \phi_n-\hat{f}\rVert_{L^2(\hat{G})}\to0$ and $\lVert \psi_n-f\rVert_{L^2(G)}\to0$
I'm able to show the first part. Observe that $\phi_n=(f1_{K_n})\text{^}$, so $\lVert \phi_n-\hat{f}\rVert_2=\lVert(f1_{K_n}-f)\text{^}\rVert_2=\lVert f1_{K_n}-f\rVert_2=\lVert f1_{K_n^c}\rVert_2 $.
Thus, $\lVert \phi_n-\hat{f}\rVert_2^2=\int\limits_{K_n^c}|f(x)|^2\ dx\to 0$ (by DCT)
But I'm unable to argue the second part i.e. $\lVert \psi_n-f\rVert_{L^2(G)}\to0$. Can anyone help me finish the proof? Thanks for help in advance.