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In page 401 of Apostol's "Mathematical Analysis"(see figure below),
the underlined statement seems to use the Intermediate value theorem to assure the existence of. But the Intermediate value theorem is applicable only when m and M are values of f for some point in S, which, however, can not be guaranteed by conditions of the theorem and the NOTE. If we additionally assume S is compact, the conclusion can hold. Whether or not there is a mistake here, or is there any other means to prove the existence of
