Stating carefully any results about countability you use, show that for any $d \geqslant 1$ the set $\mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$ of polynomials with integer coefficients in $d$ variables is countable. By taking $d=1$, deduce that there exist uncountably many transcendental numbers.

Show that there exists a sequence $x_{1}, x_{2}, \ldots$ of real numbers with the property that $f\left(x_{1}, \ldots, x_{d}\right) \neq 0$ for every $d \geqslant 1$ and for every non-zero polynomial $f \in \mathbb{Z}\left[X_{1}, \ldots, X_{d}\right]$.

[You may assume without proof that $\mathbb{R}$ is uncountable.]