Math 3751/5851 Problems

Fall 2019

  1. (Due 8/30/19) Suppose that \(A\) and \(B\) are bounded nonempty sets. Show that \(\text{sup}(A + B) = \text{sup} A + \text{sup} B\), where $$A + B = \{z : z = x + y \text{ for some } x \in A \text{ and } y \in B\}.$$

  2. (Due 9/9/19) Using the definition of the limit of a sequence, show that $$\lim \frac{n^2 - 4n + 2}{5n^2 - 3} = \frac{1}{5}. $$

  3. (Due 9/16/19) Suppose \(\ L = \lim a_n \ \) and let \(\ s_n = (a_1 + a_2 + \dots + a_n)/n.\) Show that \( \ \lim s_n = L.\)

  4. (Due 9/20/19) (a) Let \(x_1 = 2 \) and \(x_{n+1} = \sqrt{\frac{x_n + 1}{2}}\) for each \(n \in \mathbb{N}.\) Show that \(\{x_n\}\) converges.
    (b) Let \(y_1 = 1\) and \(y_{n+1} = y_n x_n\) for each \(n \in \mathbb{N}\). Show that \(\{y_n\}\) is increasing.

  5. (Due 10/4/19) (a) Show that if \(x \in E' \setminus E\), then \(x\) is a boundary point of \(E.\)
    (b) Show that \(x \notin \text{int}(E) \) if and only if there is a sequence \(\{x_n\} \subset \mathbb{R} \setminus E\) that converges to \(x\).

  6. (Due 10/9/19) (a) Suppose that \(E\) is closed and define \(\ U_n = \cup_{x \in E} (x - \frac{1}{n}, x + \frac{1}{n})\ \) for each \(n \in \mathbb{N}\). Show that \(\ E = \cap_{n=1}^\infty U_n.\)
    (b) Show that if \(U\) is an open set, then \(\ U = \cup_{n=1}^\infty F_n \ \) for some sequence \(\{ F_n\}\) of closed sets. (Think complements.)

  7. (Due 10/21/19) Using the definition of limit, show that $$\lim_{x \to 3} \frac{x+5}{4x-11} = 8.$$

  8. (Due 11/8/19) Show that if \(f\) is continuous on each closed set \(X_i\) for \(i = 1, 2, \dots, n,\) then \(f\) is continuous on \(X =\displaystyle \bigcup_{i=1}^n X_i.\)

  9. (Due 11/18/19) Show that \(\ f(x) = \sqrt{x} \ \) is uniformly continuous on \([0, \infty).\)