Show that max sup X sup Y lessthanorequalto supX Y Again do

Show that max {sup X, sup Y} lessthanorequalto sup(X Y). (Again, do not assume X is a subset of Y in this part.)

Solution

We know that X is bounded above by supX, and that Y is bounded above by supY.

We also know that supX max{supX,supY} and supY max{supX,supY} by definition of the maximum.

Thus Thus X and Y are bounded above by max{supX,supY}

Hence XUY is bounded above by max{supX,supY}

Now suppose that you have an upper bound M of XUY which satisfies M<max{supX,supY}. Since M is an upper bound for XUY , it is an upper bound for both X and Y.

Then M < sup X or M< sup Y, which is a contradiction since there cannot be a upper bound for a set which is less than its supremum.

Hence our assumption that M<max{supX,supY} is wrong. This implies that no upper bound of XUY is less than max{supX,supY}.

Hence max{supX,supY} sup(XUY)