One form of the bolzano-weierstrass theorem can be stated

Question

Recall that a point z₀ is an accumulation point of a set S if each deleted neighborhood of z₀ contains at least one point in S.

One form of the Bolzano-Weierstrass theorem can be stated as follows:

An infinite set of points lying in a closed bounded region R has at least one accumulation point in R.

Theorem 1:

Given a function f and a point z₀, suppose that:

a) f is analytic at z₀
b) f(z₀) = 0 but f(z) is not identically equal to zero in any neighborhood of z₀.

Then f(z) does not equal 0 throughout some deleted neighborhood 0<|z - z₀|< epsilon of z₀.

Using the Bolzano-Weierstrass Theorem and Theorem 1, prove that if a function f is analytic in a region R consisting of all points inside and on a simple closed contour C (except possibly poles inside C) and if all the zeros of f in R are interior to C and are of finite order, then those zeros must be finite in number.