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The book opens with the definition of the real numbers by Dedekind and goes to prove properties of this numbering system such as The supremum axiom and others.

At the end of the chapter are about 30 exercises without their solutions; some practice of the chapter and some algebraic preparation for the next chapter deals with limits (mainly inequalities).

I want to consult with you about one inequality:

Prove that if x>y>0 and n>1 natural number than

n(x-y)y

^{n-1}< x

^{n}- y

^{n}< n(x-y)x

^{n-1}

I managed to elegantly prove the left part and it goes like that:

[tex]x^{n}=[y+(x-y)]^{n}=\sum_{k=0}^{n}\binom{n}{k}y^{k}(x-y)^{n-k}=y^{n}+\sum_{k=0}^{n-1}\binom{n}{k}y^{k}(x-y)^{n-k}[/tex]

from here:

[tex]x^{n}-y^{n}=\sum_{k=0}^{n-1}\binom{n}{k}y^{k}(x-y)^{n-k}=(x-y)[\sum_{k=0}^{n-2}\binom{n}{k}y^{k}(x-y)^{n-k-1}+\binom{n}{n-1}y^{n-1}]>(x-y)ny^{n-1}[/tex]

As we were asked to show.

About the right inequality, well, it's not working so nicely...

When I try the same concept but slightly different, I have a hard time showing the last transition (where the "<" sine comes to play)

I got:

[tex]x^{n}-y^{n}=(x-y)[\sum_{k=0}^{n-2}\binom{n}{k}x^{k}(y-x)^{n-k-1}+\binom{n}{n-1}x^{n-1}][/tex]

Ans it stuck.