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→Proof: just a minor correction, D should be of radius e/2 to assure e > |w0 - w1| |
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The boundary of ''B'' is a circle and hence a [[compact set]], on which |''g''(''z'')| is a positive [[continuous function]], so the [[extreme value theorem]] guarantees the existence of a positive minimum ''e'', that is, ''e'' is the minimum of |''g''(''z'')| for ''z'' on the boundary of ''B'' and ''e'' > 0.
Denote by ''D'' the open disk around ''w''<sub>0</sub> with [[radius]] ''e/2''. By [[Rouché's theorem]], the function ''g''(''z'') = ''f''(''z'')−''w''<sub>0</sub> will have the same number of roots (counted with multiplicity) in ''B'' as ''h''(''z''):=''f''(''z'')−''w<sub>1</sub>'' for any ''w<sub>1</sub>'' in ''D''. This is because
''h''(''z'') = ''g''(''z'') + (''w''<sub>0</sub> - ''w''<sub>1</sub>), and for ''z'' on the boundary of ''B'', |''g''(''z'')| ≥ ''e'' > |''w''<sub>0</sub> - ''w''<sub>1</sub>|. Thus, for every ''w''<sub>1</sub> in ''D'', there exists at least one ''z''<sub>1</sub> in ''B'' such that ''f''(''z''<sub>1</sub>) = ''w<sub>1</sub>''. This means that the disk ''D'' is contained in ''f''(''B'').
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