Math 554 - Analysis I: Sample Final Exam Solutions - Prof. R. Sharpley, Exams of Mathematics

Download Math 554 - Analysis I: Sample Final Exam Solutions - Prof. R. Sharpley and more Exams Mathematics in PDF only on Docsity! Math 554 - Analysis I Sample Final Exam – Dec. 5, 2001 Name Directions: You must show your work for partial credit. 1. a.) State the Archimedean Principle. b.) Prove that for each
> 0 there exists n ∈ IN such that 0 < 1/n <
. 2. a.) Define the least upper bound of a set. b.) Prove that γ is the least upper bound of a nonempty set A if for each
> 0, there exists a ∈ A such that γ −
< a ≤ γ. 3. a.) Give the
− δ definition of continuity of f at x0. b.) Prove that a function f is continuous if and only if f−1[O] is relatively open for each open set O. c.) Prove that the composition of two continuous functions is continuous. (You may use any properties of continuous functions, but be certain to explain what you are doing.) 4. Show that the continuous image of a closed interval [a, b] is a closed interval, i.e. if f : [a, b] → R is continuous, then there exists a closed interval [c, d] such that f ([a, b]) = [c, d]. 5. Proved that a compact set is bounded. 6. State and sketch the proof of the Heine-Borel Theorem. 7. Suppose that f is continuous on a compact set K, then prove that f is uni- formly continuous.