# 7: Intermediate and Extreme Values

- 7.1: Completeness of the Real Number System
- Recall that in deriving the Lagrange and Cauchy forms of the remainder for Taylor series, we made use of the Extreme Value Theorem (EVT) and Intermediate Value Theorem (IVT). In Chapter 6, we produced an analytic deﬁnition of continuity that we can use to prove these theorems. To provide the rest of the necessary tools we need to explore the make-up of the real number system.

- 7.2: Proof of the Intermediate Value Theorem
- The Intermediate Value Theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval. We now have all of the tools to prove the Intermediate Value Theorem.

- 7.3: The Bolzano-Weierstrass Theorem
- The Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states that each bounded sequence in Rn has a convergent subsequence. An equivalent formulation is that a subset of Rn is sequentially compact if and only if it is closed and bounded.

- 7.4: The Supremum and the Extreme Value Theorem
- A continuous function on a closed, bounded interval must be bounded. Boundedness, in and of itself, does not ensure the existence of a maximum or minimum. We must also have a closed, bounded interval.

*Thumbnail: Bernard Bolzano. Image used with permission (Public Domain).*

### Contributors

Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)