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**Unique nth Roots in the Reals -- Rudin 1.21**

In Principles of Mathematical Analysis 1.21, Rudin sets out to show that for every positive real x there exists a unique positive nth root y. The proof is rather long and I would like to zoom into the portion of it where it seems that Rudin takes too many steps to show a sub-conclusion. This is where Rudin chooses an element h ∈ R such that the following two conditions hold:

(1) 0 < h < 1

(2) h < (x-y

^{n}) / n(y+1)

^{n-1}

He does this to say that since

(y+h)

^{n}- y

^{n}< hn(y+h)

^{n-1}< hn(y+1)

^{n-1}< x - y

^{n}

follows, we can then observe (y+h)

^{n}< x which comes from the first and last terms in the longer inequality above.

So my question/objection is as follows: why not just merely choose "h" to satisfy

(1a) 0 < h

(2a) h < (x - y

^{n}) / n(y+h)

^{n-1}

...? This way we could get to our conclusion much faster since

(y+h)

^{n}- y

^{n}< hn(y+h)

^{n-1}< x - y

^{n}

implies with fewer steps that

(y+h)

^{n}< x.