Relationship between LUB and GLB in Bounded Functions

[pessimist]

Homework Statement

h: X x Y --> R is a function from X x Y to R. X,Y nonempty. If range is bounded in R. then let

f : X --> R st f(x) = sup{h(x,y): y belongs to Y} (call this set A)
g :Y --> R st g(y) = inf{h(x,y) : x belongs to X} (call this set B)

Then prove that

sup{g(y) : y belongs to Y} is less than or equal to inf{f(x) : x belongs to X}

Homework Equations

none.

The Attempt at a Solution

As h(X,Y) is bounded the LUB and GLB exist. Now for each x, A is a subset of h(x,y). thus f(x) is <= LUB.
thus inf f(x) <= LUB.

Similarly I got, sup g(y) >= GLB.

But this leaves me nowhere :(

 

[pessimist]

well I've got it, thanks anyways.