# 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).$$