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[[Image:OpenMapping1.png|frame|right|Blue dots represent zeros of ''g''(''z''). Black spikes represent poles. The boundary of the open set ''U'' is given by the dashed line. Note that all poles are exterior to the open set. The smaller red circle is the set ''B'' constructed in the proof.]]
Assume ''f'' : ''U'' → '''C''' is a non-constant holomorphic function and ''U'' is a [[
Consider an arbitrary ''w''<sub>0</sub> in ''f''(''U''). Then there exists a point ''z''<sub>0</sub> in ''U'' such that ''w''<sub>0</sub> = ''f''(''z''<sub>0</sub>). Since ''U'' is open, we can find ''d'' > 0 such that the closed disk ''B'' around ''z''<sub>0</sub> with radius ''d'' is fully contained in ''U''. Consider the function ''g''(''z'') = ''f''(''z'')−''w''<sub>0</sub>. Note that ''z''<sub>0</sub> is a [[root of a function|root]] of the function.
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