Weierstrass factorization theorem

Remarks

The key estimate is $\abs{1 - E_p(z)} \le \abs{z}^{p+1}$ for $\abs{z} \le 1$. Choose the $p_n$ so that $\sum_n (r / \abs{a_n})^{p_n + 1} < \infty$ for every $r > 0$, for example $p_n = n - 1$. Then the Weierstrass product converges and has the prescribed zeros. For part 2, the quotient of $f$ by $z^m$ times this product is entire and has no zeros. So it is $e^{g}$ for an entire $g$, because $\C$ is simply connected. A classical example is Euler's sine product $\sin(\pi z) = \pi z \prod_{n \ge 1} (1 - z^2/n^2)$. The convergence factors are needed: for $a_n = n$ the product $\prod_n (1 - z/n)$ diverges, while $\prod_n (1 - z/n) e^{z/n}$ converges.

Going deeper

For a student

Part 1 shows that any sequence tending to infinity is the zero set of some entire function. As a consequence, every meromorphic function on $\C$ is a quotient of two entire functions: multiply it by an entire function with zeros at its poles. The theorem leaves $g$ and the $p_n$ very free. For functions of finite order the Hadamard factorization theorem makes both choices precise. The companion result for poles and principal parts is Mittag-Leffler's theorem.