Proof that $f(z)/(z-a)$ is holomorphic if $f(z)$ is holomorphic

Question

While reading the proof of Cauchy's integral formula I didn't quite understand why is $f(z)/(z-a)$; $z$ $\in$ {$\mathbb{C}$ $\neq$ $a$} holomorphic if $f(z)$ is holomorphic. I tried to prove it to myself using Cauchy-Riemann equations, but the terms didn't quite add up.

Can somebody please prove it to me using the equations or any other method?

Answer 1

Since the quotient of two holomorphic functions is holomorphic and since you are explicitely excluding $a$ from the domain, $\frac{f(z)}{z-a}$ is holomorphic too.