Prove that, for X*Y = {x*y : x in X, y in Y}, we have inf{X*Y} = inf X* inf Y and...

[esuahcdss12]

Hey all,
I just started to learn this subject and I got how to prove and to find infimum and supermium.
but I have problem with 1 question, i got now idea where to start and how to prove it.
this is the question:
Given two bounded sets X and Y , which their element are non negative REAL numbers,
so - X*Y = {x*y : x in X, y in Y},
prove that:

a) inf{X*Y} = inf X* inf Y
b)sup{X*Y} = supX *supY

Can anyone help please?

 

[stapel]

I just started to learn this subject and I got how to prove and to find infimum and supremum. But I'm having difficulty with the following question (I have no idea where to start or how to do the proof). This is the question:

Given two bounded sets, X and Y, such that all of their element are non-negative real numbers, define the set X*Y as follows:

. . . . .X*Y = {x*y : x is an element of X, y is an element of Y}

Prove the following:

. . .a) inf{X*Y} = inf X * inf Y

. . .b)sup{X*Y} = supX * supY

Okay; you say that you "got" ("understand"?) "how to find and prove" (in other words, how to work with) the infimum and supremum of a given set. But you also say that you "have no idea" how to start, let alone complete, this proof regarding the infimum and supremum of this given set. When you say that you "get" it, did you mean that you understand it when the set has specified values (that is, values much more specific that "non-negative reals"), but you're not sure how to approach this particular proof? Or do you mean something else?

Either way, when you reply, please include a clear statement of your thoughts and efforts so far. Thank you!