1. limsup and liminf

I'm trying to understand limsup and liminf and I think I'm confusing things so I was wondering if I could type out my understanding of it and have someone correct me.

The liminf of a sequence is a number above or equal to the limit that has the property such that after going down the sequence a certain number of steps beyond N no $x_n$ is greater then the limsup.

That much I think I get but I'm really not sure how this is any different then the limit since after a large enough N it will be sufficiently close to the limit. A similar problem exists for liminf for me.

Can someone show me an example where the limsup or liminf of a sequence is clearly greater or less then the limit of the sequence?

2. Originally Posted by magus
I'm trying to understand limsup and liminf and I think I'm confusing things so I was wondering if I could type out my understanding of it and have someone correct me.

The liminf of a sequence is a number above or equal to the limit that has the property such that after going down the sequence a certain number of steps beyond N no $x_n$ is greater then the limsup.

That much I think I get but I'm really not sure how this is any different then the limit since after a large enough N it will be sufficiently close to the limit. A similar problem exists for liminf for me.

Can someone show me an example where the limsup or liminf of a sequence is clearly greater or less then the limit of the sequence?
If a sequence has a limit then the limsup and the liminf of the sequence are both equal to that limit. The only way that the liminf can be different from the limsup is if the sequence fails to have a limit.

For example, the sequence given by $x_n = (-1)^n + \frac1n$ has limsup equal to 1 and liminf equal to –1.

One way to think about limsup and liminf is that the limsup of a sequence is the greatest possible limit of a subsequence, and the liminf of a sequence is the least possible limit of a subsequence. In the above example, the even-numbered terms have limit 1, and the odd-numbered terms have limit –1. If a sequence has a limit, then every subsequence has that same limit, so the limsup and liminf coincide with the limit.

3. Thank you so much. That made everything so clear. If only the book had explained it that way.