In this part, $\lambda(t) = t$ for all $t \in \mathbb{R}^+$ and $f(t) = \dfrac{\sin t}{t}$ for all $t \in \mathbb{R}^{+*}$, $f$ being extended by continuity at 0.
We denote for $n \in \mathbb{N}$ and $x \geqslant 0$,
$$f_n(x) = \int_{n\pi}^{(n+1)\pi} \frac{\sin t}{t} e^{-xt}\,dt.$$
Show that $\sum_{n \geqslant 0} f_n$ converges uniformly on $[0, +\infty[$.