(a) Using the inequality $\sin \theta \geq 2 \theta / \pi$ for $0 \leq \theta \leq \frac{\pi}{2}$, show that, if $f$ is continuous for large $|z|$, and if $f(z) \rightarrow 0$ as $z \rightarrow \infty$, then

$$\lim _{R \rightarrow \infty} \int_{\Gamma_{R}} f(z) e^{i \lambda z} d z=0 \quad \text { for } \quad \lambda>0$$

where $\Gamma_{R}=R e^{i \theta}, 0 \leq \theta \leq \pi$.

(b) By integrating an appropriate function $f(z)$ along the contour formed by the semicircles $\Gamma_{R}$ and $\Gamma_{r}$ in the upper half-plane with the segments of the real axis $[-R,-r]$ and $[r, R]$, show that

$$\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}$$