Picard's little theorem says that an entire function that omits two values is constant. The following confusion is based on a comment saying that "if an entire function omits one value, then exponential of this function will omit two values and thus constant".
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How to show that the entire function $f(z) = z^2 + \cos{z}$ has range all of $\mathbb{C}$?
Consider $f(z)=\exp(e^z)$. The function $e^z$ omits the value $0$, so $\exp(e^z)$ omits the values $0,1$. This implies $f$ is constant.
It is obvious that $f$ is not constant. So where did I go wrong?
Edit: $\exp$ takes value $1$ at $2k\pi i$.
You went wrong when you wrote that $f$ omits $1$. Take $z\in\Bbb C$ such that $e^z=2\pi i$, and then$$f(z)=\exp(2\pi i)=1.$$