Paper 4, Section II, 6E6 \mathrm{E}

Numbers and Sets | Part IA, 2007

Stating carefully any results about countability you use, show that for any d1d \geqslant 1 the set Z[X1,,Xd]\mathbb{Z}\left[X_{1}, \ldots, X_{d}\right] of polynomials with integer coefficients in dd variables is countable. By taking d=1d=1, deduce that there exist uncountably many transcendental numbers.

Show that there exists a sequence x1,x2,x_{1}, x_{2}, \ldots of real numbers with the property that f(x1,,xd)0f\left(x_{1}, \ldots, x_{d}\right) \neq 0 for every d1d \geqslant 1 and for every non-zero polynomial fZ[X1,,Xd]f \in \mathbb{Z}\left[X_{1}, \ldots, X_{d}\right].

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

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