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I am having trouble with exercise 2.4.10. I don't understand how I can prove what is being asked without using a specific range or function. Can anyone help with this?
Prof. Alekseenko: Perhaps one could start from looking at functions $ f(x)\, $ and $ g(y)\, $ closely.
Here is a hint: consider any point $ (x_0,y_0) \, $.
(1) What can be said about $ f(x_{0})\, $ and $ h(x_0,y_0) \, $?
(2) Similarly, what can be said about $ g(y_{0})\, $ and $ h(x_0,y_0)\, $?
(3) Finally, what can be said about $ f(x_0)\, $
and $ g(y_0)\, $ for any $ x_0\in X $ and $ y_0 \in Y $?
(4) How can this help to establish the desired inequality?
Can anybody fill in the detail?
Reply from a Student: So Here's my rough draft of it. I think this is enough but I could be missing something. Could not figure out how to get this looking good for the wiki so I used an image upload site. here's the link. 
Prof. Alekseenko: Thanks very much! The solution seems ok. A few minor detail can be corrected, but overall I do not see any problems with the proof. The last step, however, needs to be justified using the appropriate theorem in this section.
