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 {\bf x}_0. 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 {\bf x}_0? Thanks!