The definitions of and can get super confusing. I h0pe typing it out will help me remember it once and for all.
Let be a sequence of real numbers indexed by . Consider the sequence which is formed using the supremums of tails of the original sequence. Observe that is non-increasing (since ``the supremum over a smaller set can only get smaller''). We define.
Now if we switch the order of infimum and supremum in the above definition, the new quantity still makes sense (because the sequence of infimums of tails is non-decreasing). It is natural to define .
When faced with , think of a sequence of supremums obtained by sequentially chopping off the initial terms of the given sequence. Then take its limit, i.e., its infimum. One can interpret in a similar way.