Definition 74.
We say that a point set \(M\) is bounded if \(M\) is a subset of some closed interval.
Section 2.3 The Completeness of \(\R\)
Now we introduce a few more definitions that will lead us to the key axiom for the real numbers—completeness. We’ll continue to see interplay between sequences and sets.
Definition 75.
We say that a point set \(M\) is bounded above if there is a point \(z\) such that if \(x\in M\) then \(x\le z\text{;}\) such a point is called an upper bound.
Example 76.
The property of a point set \(M\) being bounded below and the notion of a lower bound are defined similarly; try defining them.
Example 77.
Show that a for point set (in \(\R\)), being bounded is the same as it being bounded above and below.
Example 78.
Find all upper bounds for \((0,1), [0,1]\text{,}\) and \((0,1)\cap \Q^c\) (irrationals between 0 and 1).
In the next problem, remember that we “abuse notation” by using \(\set{p_i}_{i=1}^\infty\) to mean more than one mathematical object.
Problem 79.
If the sequence \(p\) converges to the point \(x\text{,}\) then the image set \(\set{p_i}_{i=1}^\infty\) is bounded.
You can use Problem 79 to prove some of the more difficult sequence convergence properties.
Problem 80.
Show that if \(q\) converges to 0 and \(p\) converges to \(x\text{,}\) then \(\set{q_i \cdot p_i}_{i=1}^\infty\) converges to 0.
Hint.
Let \(O\) be a neighborhood of 0; why can we assume it is symmetric? Then explain why there is an \(M\) for which \(|p_i| \le M\) for all \(i\) and put the pieces together.
Now we start to approach the heart of why calculus works.
Definition 81.
We say \(p\) is a least upper bound, or supremum, of a point set \(M\) if \(p\) is an upper bound of \(M\) and \(p\le q\) for any upper bound \(q\) of \(M\text{.}\)
Example 82.
Define the greatest lower bound, or infimum, by analogy with Definition 81.
We denote the supremum of \(M\) as \(\sup(M)\) and the infimum of \(M\) as \(\inf(M)\text{,}\) when they exist.
Problem 83.
Find the suprema of \((0,1)\) and \((0,1)\cap \Q^c\text{.}\) If we could apply the definition of supremum to \(\emptyset\text{,}\) what would its supremum be?
Problem 84.
Prove that the supremum of a point set is unique, if it exists.
Problem 85.
If \(M\) and \(N\) are point sets with suprema, characterize the supremum of \(M\cup N\text{.}\)
If \(M\) and \(N\) are point sets, define
\begin{equation*}
cM := \setof{cx}{x\in M}
\end{equation*}
and
\begin{equation*}
M+N := \setof{x+y}{x\in M, \ y\in N}.
\end{equation*}
Problem 86.
Assuming \(M\) and \(N\) have suprema, prove either that \(\sup(cM) = c\sup(M)\) (if \(c \gt 0\)) or that \(\sup(M+N) = \sup(M) + \sup(N)\text{.}\)
Hint.
To show \(\sup(M+N) = \sup(M) + \sup(N)\text{,}\) let \(m = \sup(M), n= \sup(N), q = \sup(M+N)\text{,}\) and show \(m+n \le q\) and \(q \le m+n\text{.}\) For the former, it may help to prove that if \(x\le y + \epsilon\) for every \(\epsilon \gt 0\text{,}\) then \(x\le y\text{.}\)
Example 87.
Show that \(c\inf(M) = \sup(cM)\) if \(c\lt 0\text{.}\) What other properties are there relating \(\inf, \sup\text{,}\) and \(c\text{?}\)
The reason the supremum is so important is this fundamental axiom:
Axiom 88. The Completeness of \(\R\).
If \(M\) is a point set and is bounded above, then \(M\) has a supremum.
Example 89.
Come up with a sequence \(p\) such that the image set is unbounded and hence does not have a supremum.
Example 90.
Show that the axiom is not true if one requires that the supremum be a rational number.
It will be quite useful in the future to show that there is an equivalent way to formulate completeness in terms of sequences.
Definition 91.
We say that a sequence \(p\) is nondecreasing if \(p_i \le p_{i+1}\) for all \(i\in\N\text{.}\) The concept of nonincreasing is defined similarly.
Example 92.
Replace the \(\le\) above with \(\lt\) to define the notion of (strictly) increasing. Find examples of nondecreasing sequences which are not (strictly) increasing (and similarly for nonincreasing/decreasing).
Problem 93.
If \(p\) is a nondecreasing sequence such that the image set \(\set{p_i}_{i=1}^\infty\) is bounded above, then \(p\) converges to some point \(x\text{.}\)
Problem 94.
Assuming Problem 93 is true, prove the completeness axiom for point sets.
Hint.
Let \(M\) be a point set bounded above, and let \(U = \setof{u\in\R}{m\le u \forall m\in M}\text{,}\) i.e., \(U\) is the set of upper bounds of \(M\text{.}\) For every \(n\in\N\text{,}\) define \(U - \frac{1}{n} = \setof{u - \frac{1}{n}}{u\in U}\text{.}\) Show that \(U - \frac{1}{n} \cap M \ne \emptyset\text{.}\) Use this to build a nondecreasing sequence whose limit \(x\) is in \(U\text{,}\) such that \(x\le u\) for every \(u\in U\text{.}\)
Why is this all so important? One reason is that we can use the completeness of the reals to prove one of the axioms we snuck in earlier. Axiom 88 may be thought of as the “real” reason why the following is true, since open intervals can be as small as we need them to be.
Problem 95.
Using Problem 94 (but not its proof!), show that for any point \(x\text{,}\) there is an \(n\in\Z\) such that \(n\gt x\text{.}\) (Hint: contradiction!)
A use of the alternate definition of completeness is proving properties of bounded sequences. The most famous of these is the following.
The Bolzano-Weierstrauss Theorem.
Let us call a sequence \(\set{b_k}_{k=1}^\infty\) a subsequence of another sequence \(\set{a_n}_{n=1}^\infty\) if there is a sequence of natural numbers \(\set{n_i}_{i=1}^\infty\) with \(n_i \lt n_{i+1}\) such that \(b_k = a_{n_k}\text{.}\) Then every sequence with bounded image set has a convergent subsequence.
