# Homework Help: Show a function is unbounded

1. Mar 9, 2012

### k3k3

1. The problem statement, all variables and given/known data
Let f be the function defined f(x)=1/x. Prove that f is not bounded on (0,1)

2. Relevant equations

3. The attempt at a solution

I think I should prove by contradiction.

Assume f is bounded on (0,1).
Since f is bounded, there exists a real number M such that |f(x)| ≤ M for all x in (0,1)
f(x) will never be negative since it is on the interval (0,1), hence |f(x)| = f(x)

This is where I begin to get unclear on where to go next. I want to show that M+1 ≤ M
Is it correct to use 1/(M+1) and plug it into f(x)?

2. Mar 9, 2012

### Dick

I don't think you should prove it by contradiction. If n is an number greater than one then 1/n is in (0,1).

Last edited: Mar 9, 2012
3. Mar 9, 2012

### k3k3

Can I argue that since 1/n is an infinite sequence, then this function is not bounded?

4. Mar 9, 2012

### Dick

You need a better argument than that. What is f(1/n)?

5. Mar 9, 2012

### k3k3

f(1/n)=n

Then I could say for all n in the positive integers?

6. Mar 9, 2012

### Dick

You could say that, but it doesn't prove f is unbounded until you say why that proves f is unbounded.

7. Mar 9, 2012

### k3k3

Since f(1/n)=n for all n in N. Since N has an infinite amount of elements, then the function is unbounded on (0,1)?

8. Mar 9, 2012

### Dick

Having an infinite number of elements has little to do with being unbounded. What does unbounded mean?

9. Mar 9, 2012

### k3k3

That there is no lower bound, no upper bound or both.

10. Mar 9, 2012

### Dick

Ok, so give me an argument that f has no upper bound.

11. Mar 9, 2012

### k3k3

There is no n such that 1/n is not in the interval (0,1), so there is no real number M that will satisfy |1/n|≤M.

12. Mar 9, 2012

### Dick

You don't want to satisfy |1/n|<=M. You want to show that you can find a number in x in (0,1) such that f(x)>M.

13. Mar 9, 2012

### k3k3

So if M is greater than one, 1/(M+1) is in (0,1) and M+1 < M is not true?

14. Mar 9, 2012

### Dick

I really hope you meant f(1/(M+1))=M+1 > M.

15. Mar 9, 2012

### k3k3

16. Mar 9, 2012

### Dick

That's ok. But you've got it now, yes?

17. Mar 9, 2012

### k3k3

Yep. Thank you again for your help!