I think the lecturer gave you a wrong example. His example could have been and .
1. You seem to be getting the idea. When the ordinary limit exists, the limit inferior and limit superior are both equal to it.
2. As for you second question, you confusion may come from the fact that you are dealing with infinite sets. View the supremum (infimum) as a bound for and the maximum (minimum) as an element of the set. You correctly pointed out that the infimum of is 0 but 0 is not an element of which has no minimum indeed.