Question: Find$$\lim _{A \rightarrow \infty} \int_{-A}^{A} \frac{\sin(\lambda t)}{t}e^{itx}dt$$for $\lambda >0$.
My attempt: Let $f_{\lambda}=\chi_{[-\lambda,\lambda]}$. Then $$\hat {f_{\lambda}}(t)=\sqrt{\frac {2}{\pi}}\frac{\sin \lambda t}{t}$$ where the hat denotes fourier transform. Since $f_\lambda $,$\hat {f_\lambda} \in L^2$, by the Plancherel Theorem, $$||\int_{-A}^{A}\frac{\sin\lambda t}{t}e^{itx}dt-\frac{1}{\pi}f_\lambda (x)||_2\rightarrow 0$$ as $A \rightarrow \infty$.
Hence we may choose a subsequence $(A_n)_{n\in \mathbb N}$ increasing to $\infty$ such that $\int_{-A}^{A}\frac{\sin\lambda t}{t}e^{itx}dt$ converges to $\frac {1}{\pi}f(x)$ a.e. along $(A_n)_{n \in \mathbb N}$. Therefore, $\int_{-A}^{A}\frac{\sin\lambda t}{t}e^{itx}dt$ converges a.e. to $1/\pi$ if $x \in [-\lambda,\lambda]$ and a.e. to $0$ if $x \notin [-\lambda, \lambda]$.
Is this solution legitimate? Or had I made any mistake? Thanks and regards.