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

In summary, the conversation discusses a question proposed about the Riemann-Stieltjes integral in chapter 6 of "Baby Rudin." The question asks if there is a positive real number η for any given ε, such that |x|η<ε for all real numbers x. The solution proposed involves assuming that such an η does not exist, leading to a contradiction. However, a counterexample is provided that shows the statement is false.
  • #1
Arian.D
101
0

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).
 
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  • #2
Arian.D said:

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.
 

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2. Can I use different methods to prove the same fact?

Yes, there are often multiple ways to prove a fact in real analysis. It is important to choose a method that is clear and concise, and that best fits the problem at hand.

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5. What if I get stuck or can't seem to prove the fact?

If you are having trouble proving a fact, try breaking it down into smaller steps or approaching it from a different angle. You can also seek assistance from your professor or classmates for guidance and support.

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