Math 521 midterm 1

Practice problems for the first midterm

Solutions will not be posted. However, students can post their solutions on PIAZZA, and I will read and comment on those solutions. Did someone beat you to posting a solution? Read their solution and see if you agree.
  1. If $A$ and $B$ are bounded and non empty subsets of $\R$, then show that $A\subset B \implies \sup A \leq \sup B$.
  2. Let \[ a_n = 0.\overbrace{11\dots11}^{n\;{\sf digits}} = \frac{1}{10} + \frac{1}{10^2} + \cdots + \frac{1}{10^n}. \] and let $A= \{a_1, a_2, a_3, \dots\}$. Show that $x=\sup A$ exists, and show that $9x=1$. (Rudin: see section 1.22 on page 11.)
  3. If $A\subset \R$ is closed and bounded, then show that $\sup A \in A$.
  4. Let $(X, d)$ be some metric space, $a\in X$ some point in $X$, and $r\gt0$.
    1. Show that the set $E = \{x\in X \mid d(x,a)\gt r\}$ is open.
    2. Let $F$ be some subset of $B(a, r)$, and let $p$ be a limit point of $F$. Show that $d(p, a)\le r$.
    5. (About compactness)
    1. State the definition of compactness for a subset of a metric space.
    2. Show that a compact subset $E$ of a metric space $(X,d)$ is bounded.
    6. (Archimedean property)
    1. Prove the Archimedean property of the real numbers directly from the least upper bound axiom.
    2. Show that for any pair of real numbers $x\lt y$ there is a rational number $r\in \Q$ with $x\lt r\lt y$.
  7. (Variations on the Archimedean property) For each of the following questions think if the statement follows directly from the Archimedean property (Rudin: Theorem 1.20, page 9)??
    1. Show that the set $A=\{2^n: n\in \N\}$ is unbounded, i.e. show that for every real number $x$ there is an integer $n$ with $2^n\gt x$.
    2. (similar problem) Show that the set $\{n^2 \mid n\in\N\}$ is unbounded.
    3. (similar problem) Show that the set $\{n! \mid n\in\N\}$ is unbounded.
    4. (similar problem) Show that the set $\{\sqrt{n} \mid n\in\N\}$ is unbounded.
    5. (similar problem) Let $A=\{a_1, a_2, a_3, \ldots\}$ be a set of real numbers where $a_{n+1}\ge a_n+1$ holds for all $n\in\N$. Show that $A$ is unbounded.
  8. (About open and closed sets)
    1. Show that if $A\subset X$ is open , then $A^c$ is closed.
    2. Show that if $B\subset X$ is closed , then $A^c$ is open.
    3. If $A\subset\R$ is open then there is a limit point $p$ of $A$ that lies outside of $A$. True or false? (i.e. prove the statement, or give a counterexample.)
  9. Show that $\sup A = 1$ for $A = \{\frac{n}{n+2} : n\in\N\}$.
    Same question for $B=\{2^n/(2^n+1) : n\in \N\}$.
  10. Find a subset $E\subset\R$ with exactly three limit points. (Justify your answer.)
  11. (“Is this a compact set?”)
    1. Is $\Q\cap[0,1]$ a compact subset of $\R$?
    2. Is $[0,1]\setminus\Q$ a compact subset of $\R$?
    3. Is $\Q\cap[0,1]$ a compact subset of $\Q$?
    4. Show that the set $\{\frac{n}{n+3} : n\in \N\}\cup\{1\}$ is a compact subset of $\R$.