# Thread: Am I allowed to do this?

1. ## Am I allowed to do this?

I'm trying to construct an infinite sequence that converges to zero but has a positive (nonzero) limit supremum. Here's what I'm attempting to do:

Let a_j = 1 if j = n+1 and 0 otherwise. Then the limit supremum equals (lim n -> inf) sup {a_n, a_n+1, a_n+2, ...}
= (lim n -> inf) sup {0, 1, 0, 0, ...}
= 1

Am I allowed to do that, or is that "cheating"? If so, what's a workaround that would suffice?

2. ## Re: Am I allowed to do this?

Firstly, I don't think such a sequence can possibly exist.

Secondly, I'm not sure what your sequence is... Is n fixed? Is j fixed?

3. ## Re: Am I allowed to do this?

Originally Posted by Gusbob
Firstly, I don't think such a sequence can possibly exist.
Ah OK. I didn't think so, but I really don't know how to verify this.

Secondly, I'm not sure what your sequence is... Is n fixed? Is j fixed?
IDK. I was just thinking about how to make what appears to be an impossible condition possible.

4. ## Re: Am I allowed to do this?

Originally Posted by phys251
Ah OK. I didn't think so, but I really don't know how to verify this.
Check:

A sequence $\displaystyle a_n$ converges if and only if

$\displaystyle \liminf a_n = \lim a_n =\limsup a_n$

This should be obvious by definition.

5. ## Re: Am I allowed to do this?

Originally Posted by Gusbob
Check:

A sequence $\displaystyle a_n$ converges if and only if

$\displaystyle \liminf a_n = \lim a_n =\limsup a_n$

This should be obvious by definition.
Ah. That's what I thought. Thanks.