Have I proved this obvious fact correctly? (Real Analysis)

[Arian.D]

Homework Statement

It's not a HW problem. I was reading baby Rudin, in chapter 6 when it talks about Riemann–Stieltjes integral, it claims that given ε>0, we could choose η>0 such that (α(b)-α(a))η<ε. I wonder why it is true. I proposed this question to myself:

Suppose that ε>0 is an arbitrarily given number. Is there a positive real number η such that for every real x we have: |x|η<ε?

The Attempt at a Solution

My solution is this: Suppose that such an η>0 doesn't exist. Therefore there exists a real number r such that for any η>0 we have: |r|η ≥ ε. Since η is now arbitrary, let's set η=1. we'll have 0<ε<|r|, but this restricts ε and that is in contradiction with the hypothesis that ε is arbitrarily given. (for example we could set ε=|r|+1 and that fails for sure).

 

[LCKurtz]

Homework Statement

It's not a HW problem. I was reading baby Rudin, in chapter 6 when it talks about Riemann–Stieltjes integral, it claims that given ε>0, we could choose η>0 such that (α(b)-α(a))η<ε. I wonder why it is true. I proposed this question to myself:

Suppose that ε>0 is an arbitrarily given number. Is there a positive real number η such that for every real x we have: |x|η<ε?

The Attempt at a Solution

My solution is this: Suppose that such an η>0 doesn't exist. Therefore there exists a real number r such that for any η>0 we have: |r|η ≥ ε.

I don't think that follows. Just do this: Given $\eta$ and $\epsilon$, pick $x>\frac \epsilon \eta$, which shows the statement is false.