I am reading through Zagier's exposition of Newman's proof of the prime number theorem and I do not understand one of his arguments when proving his so called Analytic Theorem. This theorem states the following.
Let $f(t)~(t \geqslant 0)$ be a bounded and locally integrable function and suppose that the function $$g(z)= \int_0^\infty f(t)e^{-zt}~ dt,\quad \Re(z)>0$$ extends holomorphically to $\Re(z)\geqslant0$. Then $ \int_0^\infty f(t)~dt$ exists (and equals $g(0)$).
In his proof, Zagier states:
Let $R$ be large and let $C$ be the boundary of the region $\{z\in\mathbb C \colon |z|< R, ~\Re(z)\geqslant -\delta\}$, where $\delta > 0$ is small enough (depending on $R$) so that $g(z)$ is holomorphic in and on $C$.
My question is how do we know that there exists such $\delta$? Zagier does not give a proof of this so I assume it must be straightforward but I cannot really see how.
For those interested, Zagier's paper can be found here.
Thanks in advance.