Comprehensive Examination Advanced Calculus August 1999

A passing mark will be given if 5 problems are completely solved.

1. Let (X, d) be a metric space.

(a) Define (x_{n})_{n=1}^{∞} is a Cauchy sequence in X.

(b) Give an example of a continuous real-valued function f on (0, 1) and a Cauchy sequence (x_{n})_{n=1}^{∞} in (0, 1) such that (f(x_{n}))_{n=1}^{∞} is not Cauchy.

2. Let f_{n}(x) = \frac{nx}{1 + nx}, for n = 1, 2, ….

(a) Prove that for any a > 0, (f_{n})_{n=1}^{∞} converges uniformly on [a, ∞).

(b) Prove that (f_{n})_{n=1}^{∞} does not converge uniformly on [0, ∞).

3. Suppose a, b, c, d are real numbers such that a < b and c < d. Suppose that f :

(a, b) × (c, d) → ℝ is C^{1} and \frac{∂f}{∂x}(s, t) = 0 and \frac{∂f}{∂y} (s, t) = 0 for all s ∈ (a, b), t ∈ (c, d).

Prove that f is constant on (a, b) × (c, d).

4. Consider the series

∑_{n=1}^{∞} \frac{(1 + (−1)^{n})^{n}}{n^{2}} (x − 3)^{n}.

Determine the interval of convergence including convergence at the end points. Justify your answer.

5. Let (a_{n})_{n=1}^{∞} be a sequence of real numbers and let A be the set of all real numbers c such that there is a subsequence (a_{n_{k}})_{k=1}^{∞} with lim_{k→∞} a_{n_{k}} = c. Prove or disprove each of the following.

(a) If sup{a_{n} : n ∈ ℕ} = s is finite, s ∈ A.

(b) If sup{|a_{n}| : n ∈ ℕ} < ∞, then lim inf_{n→∞} a_{n} = inf A.

(c) If A is a singleton, lim_{n→∞} a_{n} exists.

6. Let A be a subset of a metric space (M, d). Prove that the closure of A, i.e., the smallest closed set containing A, is the union of the interior of A and the boundary of A.

7. If f is a continuous real valued function on ℝ such that f′(x) exists for all x ≠ 0 and lim_{x→0} f′(x) = 3, then f′(0) exists.