Is there a surjective function from R^m to R^n where m<n ?
My guess is no but I don't know how to justify it.
No surjective function from to with ?
Actually, for all and in , and are equipotent (i.e. there exists a bijective function between them)!
Of course, if , is finite while is infinite, so there is no surjection.
Let and be two sets, Cantor-Shröder-Bernstein theorem states that if there exists an injection from to and an injection from to , then there exists a bijection between them.
So, to start, you can find an injective function from to , to prove they are equipotent, then do the same for and , and finaly find a surjective function from to , so that will prove there is a surjection from to .
But there can be a quicker way...