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Can anyone help me prove this:

Let $X$ and $Y$ be nonempty sets and $f:X\times Y\to\Bbb R$ such that $f(X\times Y)$ is bounded. Prove the following statement:

  • $\sup_{(x,y)}f(x,y)=\sup_x\sup_yf(x,y)=\sup_y\sup_xf(x,y)\;.$

thank you

Community
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kamong
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1 Answers1

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Hint: Use the fact that $a = b$ is equivalent to $a \le b$ and $b \le a$, togeher with the inequality $$f(x, y) \le \sup \limits_y f(x, y) \le \sup \limits_x \sup \limits_y f(x, y).$$

Dominik
  • 19,963
  • Seems to be correct. – Dominik Sep 01 '15 at 20:26
  • I can't decipher the words you've written, but the second equality follows by symmetry from the first one. Alternatively, you can do the same proof with the order of $x$ and $y$ exchanged. – Dominik Sep 01 '15 at 20:33
  • I can't access this picture. If you insist on posting images of your calculations, please use a reliable imagehoster like imgur. Also, it might be better if you take some time with your calculations and try to see if your proof is correct by yourself. If you have any doubts, you can start a new question where you write down your proof [preferrably using MathJax] and we check if there are any errors. The comment section is not meant for lengthy discussions. – Dominik Sep 01 '15 at 21:11