, for some
proof:
means :
, for some
Assume
for all
but :
, for some "contradiction"
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!