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## Main Question or Discussion Point

Hi!

I need some help here, please.

In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction.

In the first part of the proof, he chooses a value $h$ such that

1. $0<h<1$

2 $h<\frac{x-y^n}{n(y+1)^(n-1)}$

My question is: how does he know such a value $h$ exists?

I know this value is positive, but I am not sure how to prove is less than 1.

To add some context:

x is a real positive number

n is a integer positive number

y is a real positive number

If someone can help me with this, I will be very grateful!

Thanks

I need some help here, please.

In Principles of Mathematical Analysis, Rudin prove the uniqueness of the n-root on the real numbers. For doing so, he prove that both $y^n < 0$ and that $y^n > 0$ lead to a contradiction.

In the first part of the proof, he chooses a value $h$ such that

1. $0<h<1$

2 $h<\frac{x-y^n}{n(y+1)^(n-1)}$

My question is: how does he know such a value $h$ exists?

I know this value is positive, but I am not sure how to prove is less than 1.

To add some context:

x is a real positive number

n is a integer positive number

y is a real positive number

If someone can help me with this, I will be very grateful!

Thanks