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let ${[{z_{n}}]}$ be sequence of complex numbers with no finite accumulation point, and let $[{w_{n}}]$ be an arbitrary sequence of complex numbers. Prove there is an entire function $f$ such that$f(z_{n})=w_{n}$ for all $n>=1$.

I have a feeling that this is an application of some theorems like Runge's Theorem and Little Picard theorem to which I am totally unconfortable. I was wondering if someone has better way of doing this without referring to the standard results.

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It's an application of Mittag-Leffler's theorem and the Weierstrass factorization theorem. See e.g. Rudin, "Real and Complex Analysis", theorem 15.15.

EDIT: In order to handle the case where all but one $w_n = 0$, you're going to need something pretty close to the Weierstrass factorization theorem. And given a solution $f$ of that case where all the zeros have multiplicity $1$ and there no other zeros, if you can solve the general problem you can divide by this $f$ and have Mittag-Leffler's theorem in the case where all the poles are simple. So it really makes sense to do this in the framework of Weierstrass and Mittag-Leffler.

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    I guess mine is the first edition, copyright 1966. Theorem 13.13 is on page 260, not page 60. Theorem 15.15 is on page 298.2012-12-21