**documents**:

Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)

l’Hospital’s: when to use- when not to use

***Review for the final (follow W. Rudin, Principles of Mathematical Analysis, 3^{rd} Ed., McGraw-Hill)***subject to change:

- know the definition of the supremum (infimum) of a set, know when this value exists, and be ready to apply
- know elementary properties of the derivative of a function.
- know the definition and properties of the closure of a set (see quiz 2 along with related proofs in Rudin)
- prove Theorem 2.37 and deduce that every bounded sequence has a convergent subsequence
- know the statement of the Heine-Borel Theorem (Theorem 2.41) and be ready to apply
- prove the Intermediate Value Theorem (see Theorem 3.15 on this sheet for a simple proof of the I.V.T.)
- know the statement of the Mean value theorem and be ready to apply
- know Minkowski’s inequality (See HW Set 10 solutions, first page)
- prove Theorem 4.8 (a useful characterization for continuity)
- know when the inverse image of an open set is open, when the image of a compact set is compact, when the extreme value theorem (Theorem 4.16) holds, and demonstrate related counter examples
- statement of Theorem 6.6 (criteria for Riemann integrability) and related definitions
- statement of Theorem 6.17 and how to use it to prove the F.T.C. (See HW Set 10 solutions, 6.5:1)
- statement of Theorem 7.12 (the uniform limit of continuous functions is continuous)
- examine the necessity of the hypotheses of Theorem 7.16
- know the statement of Stone-Weierstrass and how to apply it
- know related counter examples: a differentiable function whose derivative has a discontinuity, a function which is not continuous, but is the limit of continuous functions
- prove the Weirstrass Theorem (every bounded infinite subset of R^d has a limit point in R^d)
- be familiar with concepts presented in class on the last week (we will discuss equicontinous families of functions)
- here are some extra practice problems: practice final problems . You may also look at the problems worked in recitation for practice.