Essential singularity of e^{1/z}, function takes every value

Question

The function $e^{1/z}$ has an essential singularity in $0$ an takes every non-zero value in every neighborhood of $0$. This follows from the Picard theorem. The question is: how can we prove this without the Picard, only with Casorati-Weierstraß and open mapping theorem? Does somebody have any ideas? Thanks.