# Showing the existence of an entire function

Let $a_1, a_2, a_3 \ldots$ be a sequence of distinct complex numbers with $a_n \rightarrow \infty$. Let $b_1, b_2, b_3 \ldots$ be an arbitrary sequence of complex numbers. Show that there exists an entire function $f : \mathbb{C} \rightarrow \mathbb{C}$ such that $f(a_n) = b_n$ for all $n$.

• I was thinking of constructing a function like $f(z)=\sum g_n(z)$ where $g_n(z)$ is a polynomial; $g_n(a_i)=0$ for all $i<n$ and $g_n(a_n) = b_n - \sum_{i=1}^{n-1}g_i(a_n)$. – debanjana Jan 26 '17 at 20:37
• – Martin R Jan 26 '17 at 21:09
• @debanjana It would be good to put your comment in the question, maybe as the second paragraph. – zhw. Jan 27 '17 at 6:37