{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# QR - Q R Here follows one route from the rationals Q to the...

This preview shows pages 1–2. Sign up to view the full content.

Q R Here follows one route from the rationals Q to the reals R . The basic idea is to fashion each real number from the rational sequences that ‘converge’ to it; without having the actual real number in hand, we recognize ‘convergence’ from the Cauchy condition. Let R denote the set of all Cauchy sequences r = ( r n ) n =0 in Q ; here, to say that r is Cauchy is to say that for each ( rational !) ε > 0 there exists N such that q, p > N ⇒ | r q - r p | < ε ; such sequences are certainly bounded. Define addition and multiplication of rational sequences term-by-term: with the obvious notation, a + b = ( a n + b n ) n =0 , ab = ( a n b n ) n =0 . It is easy to check that these are binary operations that make R into a com- mutative ring, with additive identity and multiplicative identity the obvious constant sequences 0 and 1 (having constant terms 0 and 1 respectively). Let Z ⊂ R denote the set of all rational sequences z = ( z n ) n =0 that converge to zero (call them ’null’). It is easy to check that Z is an ideal in R : that is, Z

This preview has intentionally blurred sections. Sign up to view the full version.

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

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern