Deduce from hadamards theorem that if f is entire and of

Question

1. Show that the equation e^z - z = 0 has infinitely many solutions in C.

[Hint: Apply Hadamard's theorem.]

2. Deduce from Hadamard's theorem that if F is entire and of growth order ρ that is non-integral, then F has infinitely many zeros.

3. Prove that every meromorphic function in C is the quotient of two entire functions. Also, if {a_n} and {b_n} are two disjoint sequences having no finite limit points, then there exists a meromorphic function in the whole complex plane that vanishes exactly at {a_n} and has poles exactly at {b_n}.