I have proven this question but somehow I feel like maybe I have done something wrong or I have been reckless somewhere would you please help me to verify my answer
If S is a set with least upper bound and greatest lower bound property assume X and Y are nonempty subset of S such that every element of X <= every element of Y so prove Sup X <= Inf Y
I have proven this in the following way
from the hypotheses we can conclude everything in Y is an upper bound of X and vice versa.
so Sup X and Inf Y exists in s and lets denote them by a and b now lets prove this by contradiction lets suppose a>b since b is infimum of Y everything greater than b is not lower bound and this is in contradiction of a not being a lower bound of Y because a is supremum of X and since everything in Y is an upper bound of X . is my proof true or I am missing something in somewhere?