Is the set of all pairs of real numbers uncountable?

Question

My hypothesis is that $\mathbb{R \times R}$, the set of all pairs $(r_1, r_2)$, of real numbers is uncountable. I understand that the set of all pairs of natural numbers is countable. But could someone explain why the set of all pairs of real numbers uncountable? I am having trouble proving it using diagolization

Answer 1

Notice that $\Bbb R \times \{0\} \subset \Bbb R \times \Bbb R$ has the same quantity of elements that $\Bbb R$¹. So if you prove that $\Bbb R$ is uncountable, you're done.

¹ Actually $|\Bbb R| = |\Bbb R^n|$ for every $n$.

Answer 2

Real numbers are uncountable, hence the Cartesian product of real numbers is uncountable.