Principles of Analysis

Extra Credit Problems

While you are allowed to attempt these problems working with other classmates, you are NOT allowed to ask any faculty or students outside the course for help. As usual, the solutions you turn in should be your own work. Solutions may be turned in up until the end of reading period, Thursday May 2nd.

(1) Given any sequence {x_n}. Let S be the set of all possible subsequences of {x_n}. So each element of S is a particular subsequence {x_n_k}. Is S a countable or uncountable set? Provide a proof to receive credit.

(2) Prove rigorously that lim x -> 0 (sin x)/ x = 1. You may NOT use L'Hopital's rule. (We haven't proved it yet.) Try following Exercise #9 in Section 3.2.

(3) Prove the Intermediate Value Theorem using the Nested Interval Theorem. See Exercise #22 in Section 3.5.

(4) Suppose that f is a continuous function from the interval [0,1] to [0,1], that f'(x) exists and is continuous on (0,1), and that f has a fixed point q in (0,1), ie. f(q) = q. If |f'(q)| < 1, show that there exists a neighborhood U of q such that every point in U limits to q under iteration of f. In other words, given an initial seed c in U, consider the sequence {x_n} defined by x_1 = c, x_2 = f(c), x_3 = f(f(c)), ..., x_n+1 = f(x_n). Show that for any initial seed c in U, the resulting sequence always converges to q. We call such a q an attracting fixed point. Hint: Use the Mean Value Theorem repeatedly.

(5) Prove that the sum of the inifinite series of 1/n^2 from n=1 to n=inifinity is equal to Pi^2/6. You may use any reference or text to locate a proof.