In the proof of 9.14, the author just says that it is a corollary to 9.13(d). I tried using a triangular inequality, which was not successful. Also, we can know that $\hat{f} \in L^1\cap L^2,$ due to 9.13. Also, we can know that $\psi_{A}(x) \rightarrow \psi_{\infty}(x)$ as $A\rightarrow \infty$ for (almost all) $x$ by the dominated convergence theorem, because $\hat{f}\in L^1.$ But I am not sure whether $||\psi_{A}- \psi_{\infty}||_2 \rightarrow 0$ as $A \rightarrow \infty$, because for now I cannot find a dominating function that is integrable. The proof is complete upon finding such a function. But I am not sure whether this is the way that the author intended.
Since $(\psi_N)_{N=1}^{\infty}$ is a Cauchy sequence in $L^2$ with limit $f$, there exists a subsequence $(\psi_{N_k})_{k=1}^{\infty}$ whose pointwise limit is $f$ almost everywhere. Then, since $\hat{f}\in L^1$, invoking the dominated convergence theorem yields the desired result.