In part 3 of his 1885 paper, Weierstrass proved the theorem, which in the form stated by him is: if each of z_{1}, … , z_{n} is algebraic and distinct, and A_{1}, … , A_{n} are algebraic then

cannot be zero unless all of A_{1}, … , A_{n} are zero.

In the attached pdf (revised on 6 Aug 2013 – I am grateful to Fang Sun for pointing out a problem with the original which is addressed in this revision), I reproduce part 3 of Weierstrassâ€™ paper, with some minor modifications and some added explanation.

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