supremum (original) (raw)

Let A be a set with a partial order ⩽, and let X⊆A. Then s=sup⁡X if and only if:

1. 1. For all x∈X, we have x⩽s (i.e. s is an upper bound).
2. 1. If s′ meets condition 1, then s⩽s′ (s is the least upper bound).

There is another useful definition which works if A=ℝ with ⩽ the usual order on ℝ, supposing that s is an upper bound:

Note that it is not necessarily the case that sup⁡X∈X. Suppose X=]0,1[, then sup⁡X=1, but 1∉X.

Note also that a set may not have an upper bound at all.