Definition 1. The infimum of a set of numbers, {...}, is the greatest number, x, (in or out of the set) such that no element of the set is smaller than x.
Definition 2. -infinity is less than every other extended real number. +infinity is greater than every other extended real number.
Definition 3. In the context of mathematical analysis, a sequence usually means an infinite list (a,b,c,...), that is, one whose items can be labelled with the counting numbers: 1, 2, 3,... and so on forever. Sometimes the word sequence can mean a finite list. Tuple means a finite list, e.g. (9,3,4) is a tuple.
Differences between a set and a list. A set may be denoted by elements enclosed in curly brackets, {...}; its elements are separated by commas. A list, finite or otherwise, may be denoted by items enclosed in round brackets, (...); its items are separated by commas too. By definition, a set such as {7,7,2} = {7,2} = {2,7} = {7,2,7} = {2,2,2,2,2,2,2,7,2}. That is, order makes no difference, and when a set is expressed as a list, and the list contains more than one identical item, this list represents the same set as a list which differs from it only in that that item appears only once. But the list (7,7,2) does not equal the list (7,2) or the list (7,2,7) etc. The sequence of counting numbers (1,2,3,4,...) does not equal the sequence (1,1,2,3,...) or (2,1,3,4,...).
What is the infimum of {-infinity, -infinity, ... } = {-infinity, ...}? It's -infinity. That's the greatest number which no element of the set is less than.
What is the infimum of {+infinity,...}? Can't tell. Insufficient data, Captain. We need to know more about what other elements are in the set, what those dots represent. For example, if this set was {a | a > 110}, all extended real numbers, a, such that a > 0, then its infimum would be 0. But if this set was {+infinity, 52, -14}, then its infimum would be -14. And if the set contained -infinity, then -infinity would be its infimum.
Question. What is the infimum of {+infinity}? It's {+infinity}. Every other extended real number is less than +infinity, but all are less than +infinity.
Question. What's the infimum of the sequence, (...). Don't know. Undefined. The infimum has only been defined for a set. But if such a question comes up, a good guess will be that it means: what's the infimum of the set of items which make up the sequence.