In part 1 of his 1885 paper, Weierstrass demonstrated the following lemma, which is both a simplification and a generalisation of the method developed by Hermite to prove the transcendence of e:

Lemma: Let f(z) be a polynomial of degree n+1 with integer coefficients and whose roots z_{0}, … , z_{n} are all distinct. Then there exists a system of polynomials g_{0}(z), … , g_{n}(z) of degree not greater than n in z, and with integer coefficients, such that (i) each of the differences

(where ν,λ can take any of the values 0, 1, … , n) can be made arbitrarily small in absolute value, and (ii) the determinant whose elements are g_{ν}(z_{λ}) is non-zero.

In the attached pdf (updated on 1 July 2013), I reproduce Weierstrass’ proof of this lemma, with some minor modifications.

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