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-yn) / n(y+1)n-1 He does this to say that since (y+h)n - yn < hn(y+h)n-1 < hn(y+1)n-1 < x - yn 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 - yn) / n(y+h)n-1 ...? This way we could get to our conclusion much faster since (y+h)n - yn < hn(y+h)n-1 < x - yn implies with fewer steps that (y+h)n < x.