Math 3751/5851 Problems

Spring 2021

1. (Due 1/22/21) 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 2/3/21) 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 2/12/21) 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 2/17/21) (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 3/10/21) (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 3/19/21) (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 3/24/21) Using the definition of limit, show that $$\lim_{x \to 3} \frac{x+5}{4x-11} = 8.$$

8. (Due 4/5/21) Show that $$\ f(x) = \sqrt{x} \$$ is uniformly continuous on $$[0, \infty).$$

9. (Due 4/12/21) Suppose that $$f$$ is increasing and continuous on $$[a, b].$$ Show that $$f^{-1}$$ is increasing and continuous on $$[f(a), f(b)].$$

10. (Due 4/19/21) (a) Suppose that $$f'$$ is increasing on $$\mathbb R$$. Show that the graph of $$f$$ lies above any of its tangent lines.
(b) Suppose that $$f$$ is differentiable on $$[0, \infty)$$, $$f(0) = 0$$, and that the derivative $$f'$$ is an increasing function on $$[0, \infty)$$. Show that $$\frac{f(x)}{x} < \frac{f(y)}{y}$$ for all $$0 < x < y.$$