Assume that A and B arc nonempty bounded above and satisfy B

Assume that A and B arc nonempty, bounded above, and satisfy B A. Show sup B sup A.

Solution

the Definition of supremum is

if X is a (partially) ordered set, and S is a subset, then an element s0 is the supremumof S if and only if:

we have ,

s<= sup A for all s A (by definition of supremum)

=>

s <= sup A for all s B (since B is subset of A)

from the second point in the abve definition, we have

sup B <= supA

thus proved