Construction of the Real Numbers

[jgens]

Does anyone have or know of any good books that cover the construction of the real numbers via cauchy sequences? I would appreciate any recommendations. Thanks!

 

[Pinu7]

Rudin's Principals of Mathematical Analysis has it as an appendix in its first chapter which it constructs the R-numbers .

However, you should check out this(it doesn't use the "rational number" approach):
http://alexandria.tue.nl/repository/freearticles/597556.pdf

 

[ice109]

rudin doesn't use cauchy sequences - it uses dedekind cuts.

 

[snipez90]

Hmm, I think Spivak includes this construction as an exercise. IIRC, he guides you along so you get the basic idea, but the tricky parts such as demonstrating the existence of a multiplicative inverse is left entirely to you. If you want to just read about it, Terence Tao's lecture notes might be helpful:

http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week2.pdf

 

[Petek]

https://www.amazon.com/dp/0486458067/?tag=pfamazon01-20 by Thurston treats the construction of the reals by Cauchy sequences.

Petek

 

[Landau]

Axiomatic Set Theory by Suppes. You can browse through it with Google Books, see page 188.