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I was going through Perron method within the text and came across Lemma 1 which states that if u_{1},u_{2}are subharmonic on some domain D and satisfies Cauchy boundary conditions (for example), then so does max(u_{1},u_{2}). I am quite confused in terms of proving the statement. How would one theoretically explain this? Any help would be appreciated.

Thank you

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# Perron method: proving maximum to be subharmonic

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