Firstly, the set E is defined:(adsbygoogle = window.adsbygoogle || []).push({});

"LetEbe the set of all positive real numberstsuch thatt<^{n}x."

Later on the proof goes:

"Assumey>^{n}x. Putk=y/^{n}-xny. Then 0 <^{n-1}k<y. Ift≥y - k, we conclude that

y.^{n}-t^{n}≤ y^{n}-(y-k)^{n}< kny^{n-1}= y^{n}-x

Thust, and^{n}>xtis not a member ofE. It follows thaty - kis an upper bound ofE."

Why does it follow? Is it because the possibility thatt = y - kcombined with the fact thattmean that^{n}>xy-kalways has to be an upper bound ofE? Or is there some other reasoning?

Thanks in advance.

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# Rudin 1.21 Problem understanding proof of unique positive root to the

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