Topology: An Inquiry-Based Approach

Let $S$ be a nonempty subset of $\mathbb{R}$ that is bounded below. Let $a\in \mathbb{R}\text{,}$ and define $a+S$ to be $a+S=\{a+s\mid s\in S\}\text{.}$

Let $S$ be a nonempty subset of $\mathbb{R}\text{.}$

Let $A$ and $B$ be nonempty subsets of $\mathbb{R}$ that are bounded above and below. Let $A+B=\{a+b\mid a\in A,b\in B\}\text{.}$

In this exercise we prove the Archimedean property of the natural numbers. Note that the set of natural numbers, denoted $\mathbb{N}$ or ${\mathbb{Z}}^{+}\text{,}$ is the set of all positive integers.

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Theorem 5.4. The Archimedean Property.

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Given any real number $x\text{,}$ there exists a natural number $N$ such that $N>x\text{.}$

Let $x$ be a real number.

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:

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Theorem 5.5.

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Given real numbers $x$ and $y$ with $x>0\text{,}$ there exists a natural number $N$ such that $Nx>y\text{.}$

We can use greatest lower bounds to prove the following theorem.

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Theorem 5.7.

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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 $x<y\text{.}$ By the Archimedean property of the natural numbers (see Exercises 5 and Exercise 6), there is a positive integer $n$ such that $n(y-x)>1\text{.}$ Let $S=\{k\in \mathbb{Z}\mid k>nx\}\text{.}$

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.

Similar to Exercise 7 we can prove the following theorem.

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Theorem 5.8.

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Given any two distinct real numbers $x$ and $y\text{,}$ there is an irrational number that lies between them.

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 $\mathbb{R}$ (a subset of $\mathbb{R}$ is bounded if it is both bounded above and bounded below).