Math 314 Spring 2011 Practice Test IIIDr. HolmesMay 2, 2011The exam will be closed book, no notes.I may add some additional problems to this. The format will be similarto that of Test II (along with the intended method of grading it).1. Sequence limit proofs. Prove these directly from the definition of limitof a sequence.(a) limi→∞1√i= 0(b) if limi→∞ai= L and limi→∞bi= M, then limi→∞[ai+bi] = L+M2. Theorems about sequences.(a) Prove that if f is continuous at c and limi→∞ci= c, then limi→∞f(ci) =f(c) [assuming that each ciis in the domain of f ]. Of course youwon’t get any benefit by copying the proof from the notes: sit byyourself and see if you can reproduce it.(b) Prove that a subsequence of a convergent sequence converges tothe same limit. That is, if limi→∞ci= L and siis a strictlyincreasing sequence of natural numbers, then limi→∞csi= L.3. Theorems about least upper bounds(a) Prove that a set A which is nonempty and has a lower bound has agreatest lower bound. Use P13 and algebra (including propertiesof inequalities) only.(b) Prove that a nonincreasing sequence ciwhich is bounded belowconverges to the greatest lower bound of the set {cj| j ∈ N}.14. Interesting examples(a) Assume that the only numbers are the rational numbers. Describea function f and a closed interval [a, b] satisfying the conditions ofthe Intermediate Value Theorem but not the conclusion. Do thesame for the Extreme Value Theorem. In both cases, rememberthat a function can be continuous at each rational number but notextend to a continuous function on the reals.(b) Prove that the function1xis not uniformly continuous on the inter-val (0, 1]. [I might supply the definition of “uniformly continuous”if I asked