Let \(S\) be a nonempty subset of \(\R\) that is bounded below. Let \(a \in \R\text{,}\) and define \(a+S\) to be \(a+S = \{a+s \mid s \in S\}\text{.}\)
(a)
Explain why \(a+\inf(S)\) is a lower bound for \(a+S\text{.}\) Explain why \(a+S\) has an infimum.
(b)
Let \(b\) be a lower bound for \(a+S\text{.}\) Show that \(a + \inf(S) \geq b\text{.}\) Then explain why \(a+\inf(S) = \inf(a+S)\text{.}\)
2.
Let \(S\) be a nonempty subset of \(\R\text{.}\)
(a)
Assume that \(S\) is bounded above, and let \(t = \sup(S)\text{.}\) Show that for every \(r \lt t\text{,}\) there is a number \(s \in S\) such that \(r \lt s \leq t\text{.}\)
(b)
Assume that \(S\) is bounded below, and let \(t = \inf(S)\text{.}\) Show that for every \(r \gt t\text{,}\) there is a number \(s \in S\) such that \(t \leq s \lt r\text{.}\)
3.
Let \(A\) and \(B\) be nonempty subsets of \(\R\) that are bounded above and below. Let \(A+B = \{a+b \mid a \in A, b \in B\}\text{.}\)
(a)
Follow the steps below to show that \(\sup(A+B) = \sup(A) + \sup(B)\text{.}\)
(i)
Let \(x = \sup(A)\) and \(y = \sup(B)\text{.}\) Show that \(x+y\) is an upper bound for \(A+B\text{.}\)
(ii)
The previous part shows that \(A+B\) is bounded above and so has a supremum. Let \(z = \sup(A+B)\text{.}\) Explain why \(z \leq x+y\text{.}\)
(iii)
To show that \(z = x+y\) we have to prove that \(z\) cannot be strictly less than \(x+y\text{.}\) Suppose to the contrary that \(z \lt x+y\text{.}\) Let \(\epsilon = x+y-z\text{.}\) Use the result of Exercise 2 to arrive at a contradiction.
(b)
Prove that \(\inf(A+B) = \inf(A) + \inf(B)\text{.}\)
(c)
Prove or disprove: \(\sup(A \cup B) = \max\{\sup(A), \sup(B)\}\)
(d)
Prove or disprove: \(\inf(A \cup B) = \min\{\inf(A), \inf(B)\}\)
4.
Let \(X = C[a,b]\text{,}\) the set of continuous functions from \(\R\) to \(\R\) on an interval \([a,b]\text{.}\) Define \(d: X \times X \to \R\) by
\begin{equation*}
d(f,g) = \sup\{|f(x)-g(x)| \mid x \in [a,b]\}\text{.}
\end{equation*}
(a)
What is \(d(x^2,1-2x)\) on \([0,1]\text{?}\)
(b)
Prove that \(d\) is a metric on \(X\text{.}\) Describe in geometric terms how this metric measures the distance between functions \(f\) and \(g\text{.}\) (This metric is called the supremum metric or the uniform metric on \(X\text{.}\))
5.
In this exercise we prove the Archimedean property of the natural numbers. Note that the set of natural numbers, denoted \(\N\) or \(\Z^+\text{,}\) is the set of all positive integers.
Theorem5.4.The Archimedean Property.
Given any real number \(x\text{,}\) there exists a natural number \(N\) such that \(N \gt x\text{.}\)
Let \(x\) be a real number.
(a)
Suppose that there is no positive integer \(N\) such that \(N > x\text{.}\) Explain how we can conclude that \(\Z^+\) is bounded above.
(b)
Assuming that \(\Z^+\) is bounded above, explain why \(\Z^+\) must have a least upper bound \(M\text{.}\)
(c)
Explain why \(M\) cannot be a least upper bound for \(\Z^+\text{.}\) Explain why this proves the Archimedean property.
6.
In this exercise we prove two statements that are equivalent to the Archimedean property (see Exercise 5). One of the statements is the following theorem:
Theorem5.5.
Given real numbers \(x\) and \(y\) with \(x \gt 0\text{,}\) there exists a natural number \(N\) such that \(Nx \gt y\text{.}\)
(a)
Let \(x\) and \(y\) be real numbers with \(x \gt 0\text{.}\)
(i)
Show that if the Archimedean property is true, then so is Theorem 5.5.
(ii)
Show that if Theorem 5.5 is true, then so is the Archimedean property. Conclude that Theorem 5.5 is equivalent to the Archimedean property.
(b)
A second statement that is equivalent to the Archimedean property is the following.
Theorem5.6.
If \(x\) is a positive real number, then there exists a positive integer \(N\) such that \(\frac{1}{N} \lt x\text{.}\)
Prove that Theorem 5.6 is equivalent to the Archimedean property.
7.
We can use greatest lower bounds to prove the following theorem.
Theorem5.7.
Given any two distinct real numbers \(x\) and \(y\text{,}\) there is a rational number that lies between them.
This theorem tells us an important fact — that the rational numbers are what is called dense in the set of real numbers. We prove this theorem in this exercise. Let \(x\) and \(y\) be real numbers and assume \(x \lt y\text{.}\) By the Archimedean property of the natural numbers (see Exercises 5 and Exercise 6), there is a positive integer \(n\) such that \(n(y-x) \gt 1\text{.}\) Let \(S = \{k \in \Z \mid k \gt nx \}\text{.}\)
(a)
Show that \(S\) is bounded below in \(\R\text{.}\)
(b)
Explain why \(S\) contains an integer \(m\) such that if \(q \in \Z\) with \(q \lt m\text{,}\) then \(q \leq nx\text{.}\) It may be helpful to use the Well-Ordering Principle that states
Every subset of the integers that is bounded below contains its infimum.
(The Well-Ordering Principle is one of many axioms that are equivalent to the Principle of Mathematical Induction. These principles are taken as axioms and are assumed to be true.)
(c)
Explain why \(m \gt nx\) and \(m-1 \leq nx\text{.}\) Use these inequalities, along with \(n(y-x) \gt 1\text{,}\) to show that \(nx \lt m \lt ny\text{.}\) Then find a rational number that is strictly between \(x\) and \(y\text{.}\)
8.
Show that every open ball in \((\R^2,d_E)\) contains a point \(x = (x_1,x_2)\) with both \(x_1\) and \(x_2\) rational.
9.
We are familiar with solving the quadratic equation \(x^2-2 = 0\) to obtain the solutions \(\pm \sqrt{2}\text{.}\) But do we really know that the number \(\sqrt{2}\) exists? We address that question in this exercise and demonstrate the existence of the number \(\sqrt{2}\) using the greatest lower bound.
(a)
To begin, let \(S = \{x \in \R^+ \mid x^2 \gt 2\}\text{.}\) Explain why \(S\) must have a greatest lower bound \(m\text{.}\)
(b)
In what follows we demonstrate that \(m^2 = 2\text{,}\) which makes \(m = \sqrt{2}\text{.}\) We consider the cases \(m^2 \lt 2\) and \(m^2 \gt 2\text{.}\)
(i)
Suppose \(m^2 \lt 2\text{.}\) Show that there is a positive integer \(n\) such that
Explain how we have demonstrated the existence of \(\sqrt{2}\text{.}\)
10.
Similar to Exercise 7 we can prove the following theorem.
Theorem5.8.
Given any two distinct real numbers \(x\) and \(y\text{,}\) there is an irrational number that lies between them.
(a)
The first step is to demonstrate the existence of an irrational number. We will do that by proving that \(\sqrt{2}\) is irrational. Proceed by contradiction and assume that \(\sqrt{2}\) is a rational number. That is, \(\sqrt{2} = \frac{r}{s}\) for some positive integers \(r\) and \(s\) such that \(r\) and \(s\) have no positive common factors other than 1.
(i)
Explain why \(r^2=2s^2\text{.}\) Since \(2\) is prime, it follows that \(2\) divides \(r\text{.}\)
(ii)
Show that \(2\) divides \(s\text{.}\) Explain how this proves that \(\sqrt{2}\) is an irrational number.
(b)
Let \(x\) and \(y\) be distinct real numbers. Show that there exists an integer \(q\) and a positive integer \(N\) such that \(z=\frac{q\sqrt{2}}{2^N}\) is an irrational number between \(x\) and \(y\text{.}\)
Let \((X,d)\) be a metric space and \(A\) a nonempty subset of \(X\text{.}\) For \(x,y \in X\text{,}\) prove that \(d(x,A) \leq d(x,y) + d(y,A)\text{.}\)
12.
Prove that if \((X,d)\) is a metric space and \(B\) and \(C\) are nonempty subsets of \(X\text{,}\) then
\begin{equation*}
d(a, B \cup C) = \min\{d(a,B), d(a,C)\}
\end{equation*}
for every \(a \in X\text{.}\)
13.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why. Throughout, let \(S\) and \(T\) be bounded subsets of \(\R\) (a subset of \(\R\) is bounded if it is both bounded above and bounded below).
(a)
Any nonempty subset of \(S\) is bounded.
(b)
If \(S + T = \{s+t \mid s \in S, t \in T\}\text{,}\) then \(\sup(S + T) = \max\{\sup(S), \sup(T)\}\text{.}\)
(c)
Let \(S + T = \{s+t \mid s \in S, t \in T\}\text{,}\) then \(\inf(S + T) = \min\{\inf(S), \inf(T)\}\text{.}\)
(d)
If \(U\) is a nonempty subset of \(S\text{,}\) then \(\sup(U) \leq \sup(S)\text{.}\)
(e)
If \(U\) is a nonempty subset of \(S\text{,}\) then \(\inf(S) \leq \inf(U)\text{.}\)
(f)
If \(A\) is a subset of \(\R\) and \(x \in \R\) with \(d(x,A) = 0\text{,}\) then \(x \in A\text{.}\)
