Definition 72.
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 73.
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 74.
The property of a point set \(M\) being bounded below and the notion of a lower bound are defined similarly; try defining them.
Example 75.
Show that a for point set (in \(\R\)), being bounded is the same as it being bounded above and below.
Example 76.
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 77.
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 77 to prove some of the more difficult sequence convergence properties.
Problem 78.
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 79.
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 80.
Define the greatest lower bound, or infimum, by analogy with Definition 79.
We denote the supremum of \(M\) as \(\sup(M)\) and the infimum of \(M\) as \(\inf(M)\text{,}\) when they exist.
Problem 81.
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 82.
Prove that the supremum of a point set is unique, if it exists.
Problem 83.
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 84.
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 85.
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 86. The Completeness of \(\R\).
If \(M\) is a point set and is bounded above, then \(M\) has a supremum.
Example 87.
Come up with a sequence \(p\) such that the image set is unbounded and hence does not have a supremum.
Example 88.
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 89.
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 90.
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 91.
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 92.
Assuming Problem 91 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 86 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 93.
Using Problem 92 (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.
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