elementary set theory - Proving cardinality of the reals and the cross product of the reals

I am trying to prove that $\Bbb {R} \times \Bbb {R} \sim \Bbb {R}$ using the Cantor-Bernstein Theorem. So then that would mean that I need to prove that $|\Bbb {R}| \leq |\Bbb {R} \times \Bbb {R}|$ and $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$.

This is my work so far:

For the first part ($|\Bbb {R}| \leq |\Bbb {R} \times \Bbb {R}|$) I let $\Bbb {R} \rightarrow \Bbb {R} \times \Bbb {R}$ be defined as $f(r)=(r,r) , \forall r \in \Bbb {R}$, which is the identity function. So then also let $f(s) = (s,s), \forall s \in \Bbb {R}$, where $r \neq s$. So then since $(r,r) \neq (s,s)$, then $f(r) \neq f(s)$. Hence $f$ is injective and therefore $|\Bbb {R}| \leq |\Bbb {R} \times \Bbb {R}|$.

Is this correct so far? Am I on the right track?

I'm confused about how to go about proving the second part, that $|\Bbb {R} \times \Bbb {R}| \leq |\Bbb {R}|$. I know I need to show that there is an injection from $\Bbb {R} \times \Bbb {R} \rightarrow \Bbb {R}$ but I can't figure out how to do it. Also how would I define this function for the second part?