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

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?