supremum
The supremum of a set is its tightest possible ceiling. Among all the numbers that bound the set from above, the supremum is the smallest one, the ceiling you cannot lower even a hair without some element poking through.
Formally, a number s is the supremum of S, written sup S, if two things hold: s is an upper bound of S, and no number smaller than s is an upper bound. The second condition can be restated usefully: for every positive epsilon, some element of S exceeds s minus epsilon. So the supremum is approached arbitrarily closely from below by members of the set.
The supremum may or may not belong to the set. For the closed interval from 0 to 1 the supremum is 1, which is in the set; for the open interval from 0 to 1 the supremum is still 1, which is not in the set. When the supremum does belong to the set, it is also the maximum.
The deep fact that every nonempty set of reals bounded above actually has a supremum is the least upper bound property, the axiom that makes the reals complete. The rationals fail this: the set of rationals whose square is less than 2 has rational upper bounds but no rational least upper bound, because the natural ceiling, the square root of 2, is irrational.
sup {1 - 1/n : n = 1, 2, 3, ...} = 1. Each term 0, 1/2, 2/3, 3/4, ... stays below 1, yet the terms climb arbitrarily close to 1, so 1 is the least ceiling, even though 1 is never attained.
A supremum need not be reached by any element.
By convention sup of the empty set is minus infinity, and the supremum of a set unbounded above is plus infinity, in the extended reals. Among reals proper, a supremum exists only for nonempty bounded-above sets.