A proposition...

For all x in R, there exists a sequence a_n in Q such that the limit of a_n as n->+inf = x

This seems trivially false to me just from cardinality arguments (e.g. if you assume it's true then the cardinality of Q being less than the cardinality of R immediately implies that there are two different x in R that must have the same limit in Q), but I'll be damned if I can prove it rigorously.  Is there an easier way?

>>17
{0,1} is a set with cardinality greater than 1 and it does not have more than two elements.

I realize my post is poorly written with regards to the specification of S. More precisely, I meant:
S is a set, |S| > 1, {sequences of elements of S} is uncountable.