Sorry but i need a little context here. What is ? a number? a function? a space?
What is ?
You cant expect an accurate answer on this if you dont give enough information since in general that would not be true:
but
The fact that the product of non-empty sets is non-empty is, in fact, equivalent to the axiom of choice. Think about it, you need to choose any one element from for every to form some
The other way's easier. Note that now if with but then we'd have to have that there is some . Oops!