Lecture 9:

Suprema and Infima

Bounded sets

Definition. Let \(S\) be a nonempty subset of \(\R\). 

  1. The set \(S\) is said to be bounded above if there exists a number \(u\in\R\) such that \(s\leq u\) for all \(s\in S\). Each such number \(u\) is called an upper bound of \(S\).
  2. The set \(S\) is said to be bounded below if there exists a number \(w\in\R\) such that \(w\leq s\) for all \(s\in S\). Each such number \(w\) is called an lower bound of \(S\).
  3. A set is said to be bounded if it is both bounded above and bounded below. A set is said to be unbounded if it is not bounded.

1. For each of the following determine whether the statement is true or false.

 

a) \(\{1-x^2 : x\in\R\}\) is bounded above.

 


b) \(\{x : |x+1|\leq 2\}\) is bounded.

 


c) \(\{x : |x-3|\geq 4\}\) is bounded.

 


d) If \(A\) is unbounded, then \(A\) is not bounded above.

TRUE            FALSE

 

 

TRUE            FALSE

 

 

TRUE            FALSE

 

 

TRUE            FALSE

\(1\) is an upper bound since \(1-x^2\leq 1\) for all \(x\in\R\)

\(-4\) is a lower bound and \(2\) is an upper bound.

\(\{x : |x-3|\geq 4\}=(-\infty,-1]\cup[7,\infty)\)

\(A=(-\infty,1]\) is not bounded, since it is not bounded below, but \(1\) is an upper bound, so it is bounded above.

Suprema and infima

Definition. Let \(S\) be a nonempty subset of \(\R\). 

1. If \(S\) is bounded above, then a number \(u\) is said to be a supremum (or least upper bound) of \(S\) if it satisfies the following conditions:

  • \(u\) is an upper bound of S.
  • If \(v\) is any upper bound of \(S\), then \(u\leq v\).

2. If \(S\) is bounded below, then a number \(w\) is said to be a infimum (or greatest lower bound) of \(S\) if it satisfies the following conditions:

  • \(w\) is an lower bound of S.
  • If \(t\) is any lower bound of \(S\), then \(t\leq w\).

If a set \(S\) has a supremum or an infimum, then they are denoted

\[\sup S\qquad\text{and}\qquad\inf S,\]

respectively.

Suprema and infima

Definition. Let \(S\) be a nonempty subset of \(\R\). 

1. If \(S\) is bounded above, then a number \(u\) is said to be a supremum (or least upper bound) of \(S\) if it satisfies the following conditions:

  • \(u\) is an upper bound of S.
  • If \(v\) is any upper bound of \(S\), then \(u\leq v\).

It is very often that we will show the contrapositive of this is true, that is,

 

If \(u>v\), then \(v\) is not an upper bound of \(S\).

 

or, equivalently

 

If \(u>v\), then there is some \(s\in S\) such that \(s>v\).

1. For each of the following determine whether the statement is true or false.

 

a) If \(s=\sup A\), then \(s+1\) is an upper bound for \(A\).

 

b) If \(A\) has a supremum, then \(A\) has an infimum.


c) If \(s=\sup A\), then \(s+1\) is a supremum of \(A\).

 


d) If \(x\) and \(y\) are both lower bounds of \(A\) and \(x<y\), then \(y=\inf A\).

TRUE            FALSE

 

 

TRUE            FALSE

 

 

TRUE            FALSE

 

 

TRUE            FALSE

\(s\) is an upper bound, that is, \(s\geq a\) for all \(a\in A\), so clearly \(s+1\geq a\) for all \(a\in A\).

\((-\infty,2)\) has a supremum, but it is not bounded below, so it cannot have an infimum

\(s+1\) is an upper bound, but \(s\) is also an upper bound and \(s<s+1\).

\(-50\) and \(-262.12\) are both lower bounds for \(\N\), but neither is an infimum.

Proposition. \(\sup(-\infty,2)=2\).

Proof. First, it is clear that \(2\) is an upper bound for \((-\infty,2)\).

 

Now, assume that \(u<2\).  Note that

\[\frac{u+2}{2} <\frac{2+2}{2} = 2,\]

which shows that \(\frac{u+2}{2}\in(-\infty,2)\). Finally, we note that

\[\frac{u+2}{2}>\frac{u+u}{2} = u.\]

This shows that \(u\) is not an upper bound for \((-\infty,2)\). \(\Box\)

End of Lecture 9

Read Section 2.3