Practice problems: intervals, bounded sets and the bounds of a set

Write the following sets as an interval or as a union of intervals:

a) $A = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu|x + 2| < 4 \right\}$;

b) $B = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu|3 - x| \geq 1 \right\}$;

c) $C = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu x^{2} - x - 6 \leq 0 \right\}$;

d) $D = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu\frac{x - 1}{x + 2} \leq 0 \right\}$.

Consider the intervals $I = \lbrack - 3,5)$ and $J = (1,7\rbrack$. Find $I \cup J$, $I \cap J$, $I \smallsetminus J$, ${\mathbb{R}} \smallsetminus I$ and the length of the interval $I \cap J$.

Find, if they exist, the minimum, the maximum, the infimum and the supremum of the sets:

a) $A = \left\{ 1 - \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$;

b) $B = \left\{ \frac{( - 1)^{n}}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$;

c) $C = \left\{ \frac{1}{2^{n}}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$;

d) $D = ( - 1,3\rbrack \cup \left\{ 5 \right\}$;

e) $E = \left\{ n^{2} - 3n\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$.

Consider the set $A = \left\{ \frac{3n - 1}{n + 2}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$.

a) Show that $\frac{3n - 1}{n + 2} = 3 - \frac{7}{n + 2}$ for every $n \in {\mathbb{N}}$.

b) Find $\min A$.

c) Using the $\varepsilon$ characterization, show that $\sup A = 3$.

d) Does the set $A$ have a maximum? Justify your answer.

Show that the set $A = \left\{ \frac{x^{2} + 1}{x^{2} + x + 1}\mkern6mu \middle| \mkern6mu x \in {\mathbb{R}} \right\}$ is bounded, and find $\min A$ and $\max A$.

Hint: $y \in A \Leftrightarrow$ the equation $(y - 1)x^{2} + yx + (y - 1) = 0$ has at least one real solution.

Find $m \in {\mathbb{R}}$ for which the set $A = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu x^{2} - (m + 1)x + m \leq 0 \right\}$ is an interval of length $3$.

Decide whether the following sets are bounded and find their infimum and supremum:

a) $A = \left\{ \sqrt{n + 1} - \sqrt{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$;

b) $B = \left\{ n + \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$;

c) $C = \left\{ \sin x + \cos x\mkern6mu \middle| \mkern6mu x \in {\mathbb{R}} \right\}$.

Decide whether each statement is true or false; justify your answer (a proof or a counterexample):

a) Every bounded set of real numbers has a maximum.

b) If a set $A$ has a maximum, then $\sup A = \max A$.

c) If $\sup A \notin A$, then the set $A$ has no maximum.

d) Every nonempty subset of $\mathbb{N}$ has a minimum.

e) The set $\left\{ x \in {\mathbb{Q}}\mkern6mu \middle| \mkern6mu x^{2} < 2 \right\}$ has a supremum that belongs to $\mathbb{Q}$.

(optional) Let $A,B \subset {\mathbb{R}}$ be two nonempty bounded sets and $A + B = \left\{ a + b\mkern6mu \middle| \mkern6mu a \in A,\mkern9mu b \in B \right\}$. Show that $\sup(A + B) = \sup A + \sup B$.