Here's as far as I can get...

I know that

Then, how do I define

since

has an infinite number of terms?

So?? The definition of supremum is interesting ONLY if there's an infinite number of elements, otherwise we can always choose the largest one by observation which then would be the maximum
Moreover, I don't understand the convention of writing

because isn't

a single value, so the

is the same value?

Yes, is a single value FOR EVERY SINGLE n, and you take the infimum over all the possible n's!
The definition my book gives for

is: if

is the set of all subsequential limits of

, then

is

. I don't understand how this definition is equivalent to

.

If t is a partial limit then there's an infinite subsequent which converges to t and, thus there's an infinite number of elements of the original sequence which converge to t. don't you try some examples? Take for example In this sequence, we'll get for infinite n's that , so the lim sup is 1 and the lim inf is -1. Tonio
Does anyone understand what I'm asking? Lol. I'm just looking for a way to show that the two definitions are equivalent.