{"id":22850,"date":"2023-03-01T03:26:53","date_gmt":"2023-03-01T03:26:53","guid":{"rendered":"https:\/\/www.booksofall.com\/tw\/?post_type=product&p=22850"},"modified":"2023-03-01T03:26:53","modified_gmt":"2023-03-01T03:26:53","slug":"introduction-to-real-analysis","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/tw\/introduction-to-real-analysis\/","title":{"rendered":"Introduction to Real Analysis"},"content":{"rendered":"

Chapter 1 – The Real Numbers<\/h2>\n

1.1 The Real Number System<\/h3>\n

Having taken calculus, you know a lot about the real number system<\/a>; however, you probably do not know that all its properties follow from a few basic ones. Although we will not carry out the development of the real number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also to focus on one property, completeness, that is probably new to you.<\/p>\n

Field Properties<\/strong><\/p>\n

The real number system (which we will often call simply the reals) is first of all a set {a, b, c, ….} on which the operations of addition and multiplication are defined so that every pair of real numbers has a unique sum and product, both real numbers, with the following properties<\/a>.<\/p>\n

(A) a + b = b + a and ab = ba (commutative laws<\/a>).
\n(B) (a + b)+ c = a + (b + c) and (ab)c = a(bc) (associative laws<\/a>).
\n<\/span>(C) a(b + c) = ab + ac (distributive law<\/a>).
\n(D) There are distinct real numbers 0 and 1 such that a + 0 = a and a1 = a for all a.
\n<\/span>(E) For each a there is a real number 1\/a such that a + ( -a) = 0, and if \"\" , there is a real number 1=a such that a(1\/a)=1<\/p>\n

.<\/p>\n

The manipulative properties of the real numbers, such as the relations<\/p>\n

\"\"<\/p>\n

and<\/span><\/p>\n

\"\"<\/p>\n

all follow from (A)\u2013(E). We assume that you are familiar with these properties. A set on which two operations are defined so as to have properties (A)\u2013(E) is called a<\/p>\n

field. The real number system is by no means the only field<\/a>. The rational numbers (which are the real numbers that can be written as r = p\/q, where p and q are integers and \"\"a) also form a field under addition and multiplication. The simplest possible field consists of two elements, which we denote by 0 and 1, with addition defined by<\/p>\n

0+ 0 = 1 + 1 = 0 , 1 + 0 = 0 + 1 = 1<\/p>\n

and multiplication defined by<\/p>\n

0 . 0 = 0 . 1 = 1 . 0 = 0 , 1 . 1 = 1<\/p>\n","protected":false},"excerpt":{"rendered":"