The question asks to prove, that if x,yRand x<y then there must exist at least one z such that x<z<y.

My question is can I do the following:

suppose x,y are a subset of the real numbers with the set ranging from x<y. Then there must exist a z, an element of the above subset such that z=x+h. Then, x<z<y because if this was not the case then the set would not be bounded by x and y.

While I know the proof is informal does it work?