Does an oscillating sequence of zeros satisfies identity theorem (1st version i.e.limit point of zeros)

Question

Let f(z) be an analytic function in C. Then f is constant (i.e. identically zero) if the zeros of f in C contains the sequence an ={n,if 4 doesn't divide n & 1/n ,if 4|n}

Answer 1

$a_{4n}=\frac 1 {4n} \to 0$. So the zeros of $f$ have a limit point which implies it is a constant (which must be $0$).