elementary set theory - Do we have $|mathbb{Q}timesmathbb{Q} | = |mathbb{Q}|?$ or $|]a,b[|= |mathbb{R}|$?

I'm stuck into two proof about cardinality and countable set :

I have to prove that |$\mathbb{Q} \times \mathbb{Q}$| = |$\mathbb{Q}$|, i have a hint in my lessons which is $|\mathbb{N}|= |\mathbb{Q}|$, but I don't know how to proceed. This is what I did for the moment : Since $\mathbb{Q}$ is countable, it exist a $f: \mathbb{Q} \to \mathbb{N}$ which is a bijection. I define g: $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ by $$g(m,n)=2^m(2n+1)-1$$ which is bijective. then $g \circ (f \times f)$ is a bijection $\mathbb{Q} \times \mathbb{Q} \to \mathbb{N}$

Another question is to prove that $|]a,b[|= |\mathbb{R}|$, for $a