Let S be a non-empty subset of ℝ. The real number b is said to be an upper bound for S if s ≤ b for every s ∈ S. If S has an upper bound, then S is bounded above. Moreover, b is the supremum (or least upper bound) of S if b is an upper bound for S and no upper bound for S is less than b; this is written b = sup S. For example, if S = {0.9, 0.99, 0.999,…} then sup S = 1. Note here that 1 is not itself in the set S. Similarly, the real number c is a lower bound for S if c ≤ s for every s ∈ S. If S has a lower bound, then S is bounded below. Moreover, c is the infimum (or greatest lower bound) of S if c is a lower bound for S and no lower bound for S is greater than c; this is written c = inf S. A set is bounded if it is bounded above and below. More generally, in a metric space M, a subset S is bounded if S is contained in some ball.

Any non-empty set that is bounded above has a supremum—this is the completeness axiom of the real numbers—and likewise any non-empty set that is bounded below has an infimum.