Problem: For $n\in\mathbb N$, let $$f_n(x)=n\sin(n\vert x\vert)\quad\text{for }x\in\mathbb R.$$ Find the limit $\lim\limits_{n\to\infty}f_n(x)$ in $\mathcal D'(\mathbb R).$
My Attempt: Let $\psi\in\mathcal D(\mathbb R)$. Doing an integration by parts we have \begin{align*} \int_{-\infty}^\infty n\sin(n\vert x\vert)\psi(x)\,dx &=-\int_{-\infty}^0n\sin(nx)\psi(x)\,dx+\int_0^\infty n\sin(nx)\psi(x)\,dx\\ &=2\psi(0)-\int_{-\infty}^0\cos(nx)\psi'(x)\,dx+\int_0^\infty\cos(nx)\psi'(x)\,dx. \end{align*} By the Riemann-Lebesgue lemma $$\lim\limits_{n\to\infty}\int_{-\infty}^0\cos(nx)\psi'(x)\,dx=0\quad\text{and}\quad\lim\limits_{n\to\infty}\int_0^\infty\cos(nx)\psi'(x)\,dx=0.$$ It follows that $$\lim\limits_{n\to\infty}n\sin(n\vert x\vert)=2\delta_0\quad\text{in }\mathcal D'(\mathbb R).$$
Do you agree with my proof? If not, please let me know why.
Thank you for your time and appreciate any feedback.