Chap 1 The Number System

The real number m satisfying conditions i1 and i2 is

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: xists x ∈ S such that x > M − . Remark. It is not difficult to prove that the number M above is unique. It is also easy to deduce that if S is a non-empty set of real numbers and S is bounded below, then there is a unique real number m ∈ R satisfying the following two conditions: (I1) For every x ∈ S , the inequality x ≥ m holds. (I2) For every > 0, there exists x ∈ S such that x < m + . Definition. The real number M satisfying conditions (S1) and (S2) is called the supremum of the non-empty set S , and denoted by M = sup S . The real number m satisfying conditions (I1) and (I2) is called the infimum of the non-empty set S , and denoted by m = inf S . √ Let us now try to understand how numbers like 2 fits into this setting. Recall that there is √ no rational number which satisfies the equation x2 = 2. This means that the number that we know as 2 is not a rational number. We now want to show that it is a real number. Let S = {x ∈ R : x2 < 2}. Clearly the set S is non-empty, since 0 ∈ S . On the other hand, the se...
View Full Document

This note was uploaded on 02/01/2009 for the course MATH 2343124 taught by Professor Staff during the Fall '08 term at UCSD.

Ask a homework question - tutors are online