Math 3751/5851 Problems
Spring 2023
 (Due 1/20/2023) Suppose that \(A\) and \(B\) are bounded nonempty sets. Show that \(\text{sup}(A + B) = \text{sup} A
+ \text{sup} B\), where $$A + B = \{z : z = x + y \text{ for some } x \in A \text{ and } y \in B\}.$$
 (Due 1/30/2023) Using the definition of the limit of a sequence, show that
$$\lim \frac{n^2  4n + 2}{5n^2  3} = \frac{1}{5}. $$
 (Due 2/3/2023) Suppose \(\ L = \lim a_n \ \) and let \(\ s_n = (a_1 + a_2 + \dots + a_n)/n.\) Show that \( \ \lim s_n = L.\)
 (Due 2/20/2023) (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.
 (Due 3/3/2023) (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 \(x\).
 (Due 3/15/2023) (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.)
 (Due 3/20/2023) Using the definition of limit, show that $$\lim_{x \to 3} \frac{x+5}{4x11} = 8.$$
 (Due 4/3/2023) Show that if \(f\) is continuous on each closed set \(X_i\) for
\(i = 1, 2, \dots, n,\) then \(f\) is continuous on \(X =\displaystyle \bigcup_{i=1}^n X_i.\)
For graduate students: Show that this is not necessarily true if the number of sets \(X_i\) is infinite.
 (Due 4/10/2023) Show that \(\ f(x) = \sqrt{x} \ \) is uniformly continuous on \([0, \infty).\)

(Done in class) Suppose that \(f\) is increasing and continuous on \( [a, b]. \) Show that \(f^{1}\) is increasing
and continuous on \( [f(a), f(b)]. \)

(Due 4/21/2023) (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 \(0 < x < y.\)