Prove that negation of the continuum hypotheses implies existence of subset of R such that...

Question

Prove that the negation of the continuum hypothesis implies that there exist $A⊂R$ such that $ℵ_0<|A|<|R|$.

The negation of the hypotheses implies existence of a set B such that $ℵ_0<|B|<|R$|, but how can I create a subset of R from it?

Answer 1

Big HINT: By definition $|B|<|\Bbb R|$ means that there is an injection $f:B\to\Bbb R$.