My book claims that if a function $f(z)$ such that $z f(z) \rightarrow 0 $ uniformly whenever $z \rightarrow 0$ then the line integral along the semi-circular path is zero as $ R \rightarrow \infty$. $\lim\limits_{R \rightarrow \infty} \int_c f(z) = 0$ The same thing is claimed in here on Wikipedia. I don't understand this particular step
$ M_R:=\max_{\theta\in [0,\pi]} \bigl|g \bigl(R e^{i \theta}\bigr)\bigr| \to 0\quad \mbox{as } R \to \infty\,,\qquad(*)$ How does the Max value go to zero? And what does $e^{iaz}$ has to do with $zf(z)$ from my book.