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?

>>1
We can't really use cardinality like that, it's not a "size function" in the way you're thinking.

>>3
Surely the set of sequences of elements of Q is countable? Countable union of countable sets etc...

Are we allowed to assume decimal expansion? Even defining it will give the kind of proof >>1 wants.