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Springer Science+Business Media New York 2014

Peter R. Mercer More Calculus of a Single Variable Undergraduate Texts in Mathematics 10.1007/978-1-4939-1926-0_1

1. The Real Numbers

Peter R. Mercer 1

(1)

Mathematics Department, SUNY Buffalo State, Buffalo, NY, USA

Everything is vague to a degree you do not realize till you have tried to make it precise.

Bertrand Russell

We assume that the reader has some familiarity with the set of real numbers, which we denote by R . We review interval notation, absolute value, rational and irrational numbers, and we say a few things about sequences. The main point of this chapter is to acquaint the reader with two very important properties of R : the Increasing Bounded Sequence Property , and the Nested Interval Property .

1.1 Intervals and Absolute Value

For two real numbers a < b ,we write

along with the obvious definitions for [ a , b ) etc.

For a < b ,the distance from a to b is b a .One half of this distance is . The midpoint of the interval [ a , b ] is It satisfies

These are simple but useful observations. See Fig. .

Fig. 1.1

The distance between any two real numbers is measured via the absolute value function. For x R , the absolute value of x is given by

Here we agree to take the nonnegative square root of x 2. Therefore,

Since , we see that is the distance that x is from a . So for r >0,we have

We might say that x is within r of a . See Fig. .

Fig. 1.2

A few basic but important facts about absolute value are as follows.

Lemma 1.1.

Let x,y R . Then

(i)

(ii)

(iii)

(iv)

Proof.

For (i), For (ii), we need only observe that .For (iii),

Then taking (nonnegative) square roots, we get

For (iv), we write and apply (iii) to obtain

Now we reverse the roles of x and y ,and use (ii), to get

Taken together, these last two inequalities read

That is,

In Lemma why it gets this name. Item (iv) is also useful; it is called the reverse triangle inequality .

Remark 1.2.

The trick used in the proof of item (iv) in Lemma , of subtracting y and adding y ,then using the triangle inequality, is very common in calculus and real analysis.

1.2 Rational and Irrational Numbers

We denote by N the set of natural numbers :

The set N is closed under addition and multiplication, but it is not closed under subtractionthat is, the difference of two natural numbers need not be a natural number.

Appending to N all differences of all pairs of elements from N , we get the set of integers :

The set Z is closed under addition, multiplication, and subtraction. But Z it is not closed under divisionthat is, the quotient of two integers need not be an integer.

Appending to Z all quotients of all pairs of elements from Z (with nonzero denominators) we get the set of rational numbers :

The set Q is closed under addition, multiplication, subtraction, and division. (The reader should agree that it is indeed closed under division.) But as the following lemma shows, Q is not closed under the operation of taking square rootsthat is, the square root of a rational number need not be a rational number.

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