# Boundedness Removed

• January 24th 2010, 09:25 PM
frenchguy87
Boundedness Removed
Given the following theorem,
Thm: Let X be a bounded sequence of reals and let x have the property that every convergent subsequence of X converges to x. Then the sequence x converges to x.

Give an example to show that the theorem fails if the hypothesis that X is bounded is removed
• January 24th 2010, 10:09 PM
Drexel28
Quote:

Originally Posted by frenchguy87
Given the following theorem,
Thm: Let X be a bounded sequence of reals and let x have the property that every convergent subsequence of X converges to x. Then the sequence x converges to x.

Give an example to show that the theorem fails if the hypothesis that X is bounded is removed

What do you think? If we removed boundedness, what do you think is the obvious place to look?
• January 25th 2010, 04:08 AM
frenchguy87
I was thinking outside the original bound M
• January 25th 2010, 05:21 AM
HallsofIvy
1, 1/2, 2, 1/3, 3, 1/4, 4, 1/5, 5, ...
• January 25th 2010, 06:49 AM
frenchguy87
I'm not sure that works since every subsequence has to converge to the same limit x. I might be understanding it wrong though
• January 25th 2010, 07:03 AM
Defunkt
Quote:

Originally Posted by frenchguy87
I'm not sure that works since every subsequence has to converge to the same limit x. I might be understanding it wrong though

Every convergent subsequence will converge to the same limit. In HallsofIvy's example, any convergent subsequence will be of the form $\frac{1}{n_k}, n_k \to \infty$ starting some $N\in \mathbb{N}$, and will thus converge to 0.