Math 521 midterm 1

Practice problems for the first midterm

1. If $A$ is a nonempty bounded subset of $\R$ is it then always true that $\inf A \le \sup A$? (give a proof or a counterexample). Is it always true that $\inf A\lt\sup A$?
2. If $A$ and $B$ are bounded and non empty subsets of $\R$ with $A\subset B$.
   1. Show that $\sup A\leq \sup B$ .
   2. Is it always true that $\inf A\leq \sup B$ ? (proof or counterexample)
   3. Is it always true that $\inf B\leq \sup A$ ? (proof or counterexample)
3. Let $A\subset\R$ be a bounded set of integers. Show that $\sup A$ is an integer.
   Let $x=\sup A$. We show by contradiction that $x\in A$. Suppose $x\not\in A$. Since $x$ is the least upper bound of $A$, there is an $n\in A$ with $x-1\lt n\lt x$. The interval $(x-1, x)$ cannot contain more than one integer, so $n$ is the only number from $A$ that belongs to $(x-1,x)$. Therefore $n$ is an upper bound for $A$, even though $n\lt x$, and $x$ was the least upper bound. This is a contradiction, and we conclude that $x=\sup A$ must belong to $A$.
4. Let $A\subset\N$. Show that $A$ contains a smallest number, i.e. there is an $n\in A$ such that every $m\in A$ satisfies $m\geq n$.
5. Let $A=\{ \frac{n}{2n+1} : n\in\N\}$. Show that $\sup A = \frac12$.
   Solution. For every $n\in\N$ we have $2n+1\gt 2n$, so that \[ \frac{n}{2n+1} \lt \frac{n}{2n} = \frac{1}{2}. \] Therefore $\frac12$ is an upper bound for $A$.

   Let $x\lt \frac12$, and assume that $x$ is an upper bound for $A$. This implies that \[ \forall n\in\N: \quad \frac{n}{2n+1} \le x \] Hence \[ \forall n\in\N: \quad n \le 2nx + x, \] and thus \[ \forall n\in\N: \quad (1-2x)n \le x. \] Since $x\lt \frac12$ we have $1-2x\gt0$ so we may divide both sides of the inequality by $1-2x$ and conclude that \[ \forall n\in\N:\quad n \le \frac{x}{1-2x}. \] This implies that $x/(1-2x)$ is an upper bound for the set of natural numbers, in direct contradiction with the archimedean property.

   The contradiction shows that no number less than $\frac 12$ can be an upper bound for $A$.

   Hence $\frac{1}{2}$ is the least upper bound for $A$.

6. Show that $\sup A = 1$ for $A = \{\frac{2^n}{2^n+1} : n\in\N\}$.
7. (Archimedean property and variations)
   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 $x\lt r\lt y$.
   3. Show that the set $A=\{3^n: n\in \N\}$ is unbounded, i.e. show that for every real number $x$ there is an integer $n$ with $3^n\gt x$.
   4. Show that the set $\left\{\frac{n^2}{n+1} \mid n\in\N\right\}$ is unbounded.
   5. Show that the set $\{n! \mid n\in\N\}$ is unbounded.
   6. Show that the set $\{\sqrt{n} \mid n\in\N\}$ is unbounded.
   7. 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. Let $E=\{\frac1n : n\in\N\}$, and let $F=E\cup \{0\}$. Find all the limit points of $E$ and of $F$. Are either of $E$ or $F$ closed?
9. Let $A = \{\frac{m}{m+2}| m\in\N\}$ and $B=\{\frac{m}{m-2}|m\in\N, m\geq 3\}$. Find all limit points of the set $A$ and of the set $B$.
10. Find a subset $E\subset\R$ with exactly three limit points. (Justify your answer.)
11. Let $E\subset\R$ be a bounded subset, and let $m$ be a limit point of $E$. Show that $m\leq \sup E$.
12. 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 $d(p, a)\le r$.
13. Find all limit points of the set $\N\subset\R$.
14. (About open and closed sets) Notation: $A^c = X\setminus A = \{x\in X | x\not\in A\}$.
    1. Show that if $A\subset X$ is open , then $A^c$ is closed.
    2. Show that if $B\subset X$ is closed , then $B^c$ is open.
    3. True or false? If $A\subset\R$ is open then $A$ contains none of its limit points (i.e. prove the statement, or give a counterexample.)
    4. True or false? If $A\subset\R$ is open then there is a limit point $p$ of $A$ that lies outside of $A$.
    5. True or false? If $E\subset X$ is closed, then $E$ contains no limit points of its complement (i.e. $E$ contains no limit points of $E^c$.)
    6. True or false? If $E\subset X$ is open, then $E$ contains no limit points of its complement (i.e. $E$ contains no limit points of $E^c$.)