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 \(0 < x < y.\)