Vrftsjtryk

Theorem. Given two countably infinite sequences of complex numbers ${a_{n}}_{n}$ and ${b_{n}}_{n}$ with $lim_{n to infty}|a_{n}| = infty$, it is always possible to find a entire function $F$ that satisfies $F(a_{n}) = b_{n}$ for all $n$.

First, I want to prove

Problem. If $a_1,dots,a_n$ and $b_1,dots,b_n$ are distinct complex numbers, I can construct a polynomial $p$ of degree $leq n-1$ such that $p(a_i) = b_i$ for all $i=1,dots,n$.

My approach.

$$c_{0} + c_{1}a_{1} + c_{2}a_{1}^{2} + cdots + c_{n-1}a_{1}^{n-1} = b_{1}$$
$$c_{0} + c_{1}a_{2} + c_{2}a_{2}^{2} + cdots + c_{n-1}a_{2}^{n-1} = b_{2}$$
$$vdots$$
$$c_{0} + c_{1}a_{n} + c_{2}a_{n}^{2} + cdots + c_{n-1}a_{n}^{n-1} = b_{n}$$
and so
$$left(begin{array}{ccccc}
1 & a_{1} & a_{1}^{2} & cdots & a_{1}^{n-1}\
1 & a_{2} & a_{2}^{2} & cdots & a_{2}^{n-1}\
vdots & vdots & vdots & ddots & vdots\
1 & a_{n} & a_{n}^{2} & cdots & a_{n}^{n-1}\
end{array}right)
left(begin{array}{c}
c_{0}\
c_{1}\
vdots\
c_{n-1}
end{array}right)
=
left(begin{array}{c}
b_{1}\
b_{2}\
vdots\
b_{n}
end{array}right)$$

What is the best method to solve this system? Or, Is there a better approach to this problem??