Zeros of $e^{z}-z$, Stein-Shakarchi Complex Analysis Chapter 5, Exercise 13 This is an exercise form Stein-Shakarchi's Complex Analysis (page 155) Chapter 5, Exercise 13:

Prove that $f(z) = e^{z}-z$ has infinite many zeros in $\mathbb{C}$.

Attempt: 
If not, by Hadamard's theorem we obtain $$e^{z}-z = e^{az+b}\prod_{1}^{n}(1-\frac{z}{z_{i}})$$ where $\{z_{i}\}$ are the zeros of $f$. How can we conclude ?

Are you allowed to use Picard's Theorem?
If yes here is a relative question:
Use Picard's Theorem to prove infinite zeros for $\exp(z)+Q(z)$
