Abstract

The first chapter of the classic book Principles of Mathematical Analysis (WALTER RUDIN, Third Edition) is the real
and complex number systems. Theorem 1.20 of the chapter is extracted from the construction of real number R, and it
provides a good illustration of what one can do with the least-upper-bound property. Besides, theorem 1.21 proves the
existence of nth roots of positive real numbers. Remarks were given because of the importance of the two theorems.
Particularly, the proof of “Hence there is an integer m (with?m2 ? nx ? m1 ) such that m ? 1 ? nx < m . ”which is not
mentioned in theorem 1.20 was given in 2.2. A loophole “For every realx > 0 and every integern > 0 there is one and only
one realy such that yn = x.” and a flaw of “If t = x
1+x then 0 < t < 1. Hence tn < t < x.” of theorem 1.21 were indicated in
3.1 and 3.2.

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