We're told that $f_n(\lambda)=\int_{n}^{+\infty} e^{-\lambda t}\sin{t}\ dt$ converges uniformly to $0$ when $n\to\infty$.
Is it really true ? I can't see how to bound that with something going to $0$ when $n\to\infty$ independently of $\lambda$...
For example, we could do
$\left| \int_{n}^{+\infty} e^{-\lambda t}\sin{t}\ dt\right|\le \int_{n}^{+\infty} e^{-\lambda t}\ dt=\frac{e^{-\lambda n}}{\lambda}\xrightarrow{n\to+\infty} \ 0,$ but the convergence isn't uniform, is it ?