Question: Let $f$ be an entire function that satisfies $|f(z)| \leq 1 + |z|$ for all $z \in \mathbb{C}$. Show that $f(z) = az + b$ for fixed complex numbers $a$ and $b$.
My attempts: I realise that I need to apply the generalised Cauchy Integral Formula $\displaystyle f^{(n)}(z_0)=\frac{n!}{2\pi i} \int_{\Gamma}\frac{f(z)}{(z-z_0)^{n+1}}\,dz$ on an arbitrarily large circle to show that $f^{(n)}(0)=0$ for all $n \geq 2$. While I know what to after this is proven, I'm unsure how to actually prove this. Any help or guidance would be greatly appreciated!