For totally ordered set B, A is contained in B, A is an upper bound of A (any b∈B has b≤a), and any ε >; 0, c∈A exists, so a-ε.

The supremum of function should refer to the supremum of function image set in R.

Mathematical definition of the lower supremum

Bounded set s, if ξ satisfies the following conditions

(i) For all x∈S, x≥ξ, that is, ξ is the lower bound of s;

(ii) for any β>0, x∈S exists, which makes x

Then ξ is called the lower supremum of the set S, and it is recorded as ξ=infS.