Berge's lemma is a result in mathematical optimization and game theory that deals with the properties of set-valued functions. It states that if a set-valued function is upper hemicontinuous and compact-valued, then the maximum of this function can be achieved at some point in its domain. This lemma is particularly useful in proving the existence of equilibria in various economic models. The lemma is named after Frédéric Berge, a French mathematician who contributed significantly to the fields of optimization and game theory. It provides a foundational tool for analyzing problems where multiple outcomes or strategies are involved, ensuring that optimal solutions can be found under certain conditions.