123 Theorem 5.2.1 Let Dbe a nonempty compact subset of Rand let f : D→Rbe a continuous function. Then f (D) is a compact subset of R. In particular, f (D) is closed and bounded. Proof: Take any sequence {yn}in f (D). Then for each n, there exists an ∈Dsuch that yn =f (an). Since Dis compact, there exists a subsequence {ank}of {an}and a point a∈Dsuch that lim k→∞ ank =a∈D. It now follows from Theorem 3.3.2 that lim k→∞ ynk =lim k→∞ f (ank) =f (a) ∈ f (D). Therefore, f (D) is compact. The final conclusion follows from Theorem 5.1.5□ We now prove the Extreme Value Theorem in our more general context. Theorem 5.2.2 — Extreme Value Theorem. Suppose f : D→Ris continuous and Dis a compact set. Then f has an absolute minimum and an absolute maximum on D. Proof: Since Dis compact, A=f (D) is closed and bounded (see Theorem 5.1.5). Let m=infA=inf x∈D f (x). In particular, m∈R. For every n∈N, there exists an ∈Asuch that m≤an <m+1/n. For each n, since an ∈A=f (D), there exists xn ∈Dsuch that an =f (xn) and, hence, m≤f (xn) <m+1/n. By the compactness of D, there exists an element x0 ∈Dand a subsequence {xnk}that converges to x0 ∈Das k →∞. Because m≤f (xnk) <m+ 1 nk for every k, by the squeeze theorem (Theorem 2.1.3) we conclude limk→∞ f (xnk) =m. On the other hand, by continuity we have limk→∞ f (xnk) = f (x0). We conclude that f (x0) =m≤ f (x) for every x ∈D. Thus, f has an absolute minimum at x0. The proof is similar for the case of absolute maximum. □ Remark 5.2.1 The proof of Theorem 5.2.2 can be shortened by applying Theorem 5.1.4. However, we have provided a direct proof instead. The version of the Extreme Value Theorem presented in Chapter 3 is now a simple corollary of the more general version. Corollary 5.2.3 If f : [a,b] →Ris continuous, then it has an absolute minimum and an absolute maximum on[a,b]. Corollary 5.2.3 follows immediately from Theorem 5.2.2, and the fact that the interval [a,b] is compact (see Example 5.1.4). The following theorem shows one important case in which continuity implies uniform continuity.
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