### Remarks about Two Theorems in Principles of Mathematical Analysis

#### 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.

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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Journal of Mathematics Research ISSN 1916-9795 (Print) ISSN 1916-9809 (Online)

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