I am stuck with the following problem: (GATE-Question)
Let $f:\mathbb C\rightarrow \mathbb C$ be an arbitrary analytic function satisfying $f(0)=0$ and $f(1)=2.$ Then, which of the following items is correct?
(a) there exists a sequence $\{z_{n}\}$ such that $|z_{n}|> n$ and $|f(z_{n})|> n$,
(b) there exists a sequence $\{z_{n}\}$ such that $|z_{n}|>n$ and $|f(z_{n})|< n$,
(c) there exists a bounded sequence $\{z_{n}\}$ such that $|f(z_{n})|> n$,
(d) there exists a sequence $\{z_{n}\}$ such that $z_{n} \rightarrow 0$ and $f(z_{n})\rightarrow 2.$
I do not know how to progress with the problem or what property to use. Please help. Thanks in advance for your time.