1549901369-Elements_of_Real_Analysis__Denlinger_

2 Chapter 1 • The Real Number System

efficient and honest. The constructive approach would detain us too long , and

the pragmatic approach is neither rigorous nor intellectually honest. Readers

interested in the constructive approach may consult references^1 [10], [11], [25],

[37], [44], [63], [74], [82], [96], [117], and [129] listed in the Bibliography.

Much of the methodology in Sections 1.1-1.4 is not really characteristic of

analysis. It more closely resembles algebra than analysis. This material appears

here because it provides us with a rigorous starting point. You are perhaps

already familiar with most of these results, having covered them in a course

in proof theory or abstract algebra. You may only need to review such results

briefly. But the methods used in the proofs here are important enough for you

to pay careful attention to, at least once in your life. Remember, we are laying

a foundation that must be secure enough to allow us to prove all t he results of

calculus.

LOGICAL SYMBOLS USED IN THIS TEXT

Certain principles of logic are used frequently in real analysis. I have found
that students who recognize and understand these principles learn the subject
more easily than those who don't. To help the reader be aware of the presence
of important logical patterns we use standard logical symbols. In particular, we
sh all make frequent use of the following symbols:

• Implication: P ==> Q
  If P and Q are statements, then "P ==> Q " is the statement "P implies
  Q ," which is equivalent to each of the following:
  if P, then Q if P, Q
  Q if P P only if Q


• Bi-implication: P {:::} Q (often written "P iff Q")
  If P and Q are statements, then "P {:::} Q" is the statement "P if and
  only if Q," which is equivalent to the conjunction "P ==> Q and Q ==> P."


• Universal Quantifier: '<Ix EA, P(x)
  If P(x) is a statement about x, and A is a set, t hen "'<Ix E A, P(x)"
  is the statement, "For all x in the set A, P(x) is true." If t he set A is
  understood without writing it, we sometimes write simply "'<Ix, P(x) ."
  Variations of this usage will b e self-explanatory when we use them.


• Existential Quantifier: 3 x EA 3 P(x)
  If P(x) is a statement about x, and A is a set, then "3 x EA 3 P (x)" is
  the statement, "There exists (at least one) x in the set A such t h a t P(x)
  is true." Variations of this usage will be self-explan atory.

1. The numbers in square brackets refer to entries in the Bibliography, which follows Appendix
   c.