= Algebraically independent real sequence
{title2=$f(x_1,\ldots,x_d)\ne0$}
There is a sequence of <real numbers> for which every finite initial segment consists of <algebraically independent elements> over the rationals. Given an independent initial segment, specialize each nonzero integer polynomial in one extra variable at those values. At least one coefficient remains nonzero, so each resulting polynomial has finitely many roots. There are countably many such polynomials by <countability of integer polynomial rings>. Choosing the next real outside the countable union of their roots continues the construction. Clearing denominators makes independence over the integers equivalent to independence over the rationals.
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