Answer to Question #3985 in Real Analysis for Junel

Question

If f(x)<=g(y) for all x,y element D, then we may conclude that the upremum f(D)<=infimum g(D), which we may write as: Sup f(x)<=inf g(y) such that x,y element of D

Answer 1

Let's prove the statement.Suppose that the statement is wrong, then supremum f(D) > infinum g(D).
Then there exists such x0 - element of D, that f(x0) <= supremum f(D) and
at the same time f(x0) >infinum g(D). As we know, f(x) <= g(y) for all x, y from D.
That's why
f(x0) <= g(y) for all y from D. But this statement proves that f(x0) is infinum g(D).
We came to the contradiction because f(x0) > infinum g(D).
So, out assumption was wrong and supremum f(D) <= infinum g(D).