 Solution

The Liouville theorem states that every bounded entire function is constant. Let $|f(z)|\le M$ and choose distinct $a,b$. For $R>\max (|a|,|b|)$, partial fractions and the Cauchy integral formula givewith the circle traversed anticlockwise. On that circle,The displayed difference is independent of $R$, so $f(a)=f(b)$. Since $a,b$ were arbitrary, $f$ is constant. This proves the theorem with the requested two-pole contour rather than assuming it.