Worksheet

Worksheet: intervals, bounded sets and the bounds of a set

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Remember

  • $M$ is an upper bound of $A$ if $x \leq M,\mkern9mu\forall x \in A$; $m$ is a lower bound of $A$ if $m \leq x,\mkern9mu\forall x \in A$.
  • $\sup A$ = the least upper bound of $A$; $\inf A$ = the greatest lower bound of $A$.
  • $s = \sup A \Leftrightarrow$ (1) $x \leq s,\mkern9mu\forall x \in A$ and (2) $\forall\varepsilon > 0,\mkern9mu\exists x_{\varepsilon} \in A$ such that $x_{\varepsilon} > s - \varepsilon$.
  • $i = \inf A \Leftrightarrow$ (1) $x \geq i,\mkern9mu\forall x \in A$ and (2) $\forall\varepsilon > 0,\mkern9mu\exists x_{\varepsilon} \in A$ such that $x_{\varepsilon} < i + \varepsilon$.
  • $A$ has a maximum $\Leftrightarrow \sup A \in A$, and then $\max A = \sup A$. If $A$ is not bounded above, $\sup A = + \infty$; if $A$ is not bounded below, $\inf A = - \infty$.

  1. Write the following sets as an interval or as a union of intervals, and say which of them are bounded:

    a) $A = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu|x - 1| \leq 3 \right\}$;

    b) $B = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu x^{2} - 4x + 3 < 0 \right\}$;

    c) $C = \left\{ x \in {\mathbb{R}}\mkern6mu \middle| \mkern6mu|2x + 1| > 5 \right\}$.

  2. Fill in the table (write “yes”/“no”, the value asked for, or “does not exist”):

    The set bounded above? bounded below? min max inf sup
    $A = \lbrack - 2,5)$
    $B = \left\{ \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$
    $C = \left\{ ( - 1)^{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}} \right\}$
    $D = (0, + \infty)$
    $E = {\mathbb{Z}} \cap ( - 3,2\rbrack$

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

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

    b) Show that $2 < x \leq 3$ for every $x \in A$.

    c) Find $\max A$.

    d) Using the $\varepsilon$ characterization, show that $\inf A = 2$. Does the set $A$ have a minimum?

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

    Hint: $x^{2} + 1 \geq 2|x|$ for every $x \in {\mathbb{R}}$.

  5. (optional) Find $\inf A$ and $\sup A$ for $A = \left\{ ( - 1)^{n} + \frac{1}{n}\mkern6mu \middle| \mkern6mu n \in {\mathbb{N}}^{*} \right\}$. Say whether the set $A$ has a minimum and whether it has a maximum.