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If $f$ is non-constant and entire, prove that there exists a $z_0 \in C$ such that $f(z_0)$ is a positive real number, without appealing to Picard's theorem.

The obvious approach to this would be to modify a proof of Picard's little theorem to try and prove this, but all of the proofs I have found of little Picard's seem to be higher level than I have - I'm trying to solve this using theorems from a quarter long course in complex. I would really appreciate a hint in the right direction....

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If not, $1 - f(z)$ never takes a value in $(-\infty, 0]$, and so there is an analytic branch $g$ of $\sqrt{1 - f(z)}$ with real parts always positive. What can you say about $\exp(-g)$?

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