Let f be an entire function such that for every integer k ≥ 1 there is an infinite set Xk such that f(z) = \(\rm \frac{1}{k}\) for all z ∈ Xk.. Which of the following statements are necessarily true?
1
There exists an infinite set X such that f(z) = 0 for all z ∈ X
2
There exists a non-empty closed set X such that f(z) = 0 for all z ∈ X
3
The set Xk is unbounded for each k ≥ 1
4
If there exists a bounded sequence (zk)k≥1 such that zk ∈ Xk for each k ≥ 1, then f has a zero