Question

# Let S be an ordered set and A is a nonempty subset such that sup A exists. Suppose there is a B⊂A such that whenever x∈A there is a y∈B such that x≤y. Show that supB exists and supB=supA.

Analyzing functions
Let S be an ordered set and A is a nonempty subset such that sup A exists. Suppose there is a $$\displaystyle{B}⊂{A}$$ such that whenever $$\displaystyle{x}∈{A}$$ there is a $$\displaystyle{y}∈{B}$$ such that $$\displaystyle{x}≤{y}$$. Show that $$\displaystyle\supset{B}$$ exists and $$\displaystyle\supset{B}=\supset{A}$$.