Notes - UCSD Math Department
... Claim. S is nonempty and bounded from above.
Proof of claim. (1) By Theorem 1.20(a), since 1 > 0 and x 2 R, there
exists a positive integer n such that n = n 1 > x. That is, n < x; which
says that n 2 S. So S is nonempty.
(2). By Theorem 1.20(a) again, since 1 > 0 and x 2 R, there exists a positive