A function $f(z)$ is analytic within a closed contour $C$ (and continuous on $C$). If $f(z)\neq 0$ within $C$ and $|f(z)|\leq M$ on $C$, show that 
$|f(z)|\leq M$ for all points within $C$.

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> Hint: Consider $w(z)=\frac{1}{f(z)}$
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I am unable to make any progress in this question, it lies under the section of Cauchy Integral Formula. 
Any help is appreciated. Thanks.