- #1
jmjlt88
- 96
- 0
Let X and Y be nonempty sets and let h: X x Y→ℝ
Define f: X →ℝ and g: Y→ℝ by the following:
f(x)=sup{h(x,y): y in Y} and g(y) = inf{h(x,y): x in X}
Prove that sup{g(y): y in Y} ≤ inf{f(x): x in X}
Attempt at solution:
Pick y' in Y. Then g(y')≤h(x,y') for all x in X. Hence, there exist some x' such that g(y')≤h(x',y').
Then, h(x',y')≤ sup{h(x',y): y in Y} = f(x')... Not too sure where to go from here... A small hint would be great!
Thanks!
Define f: X →ℝ and g: Y→ℝ by the following:
f(x)=sup{h(x,y): y in Y} and g(y) = inf{h(x,y): x in X}
Prove that sup{g(y): y in Y} ≤ inf{f(x): x in X}
Attempt at solution:
Pick y' in Y. Then g(y')≤h(x,y') for all x in X. Hence, there exist some x' such that g(y')≤h(x',y').
Then, h(x',y')≤ sup{h(x',y): y in Y} = f(x')... Not too sure where to go from here... A small hint would be great!
Thanks!