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We provide a new proof that the subdifferential of a proper lowersemicontinuous convex function on a Banach space is maximal monotone byadapting the pattern commonly used in the Hilbert setting. We then extend thearguments to show more precisely that subdifferentials of proper lowersemicontinuous prox-bounded functions possess the Br{\o}ndsted-Rockafellarproperty.

Br{\o}ndsted-Rockafellar property of subdifferentials of prox-bounded functions