Say we try to define the Bernoulli numbers using Faulhaber's formula...

$\displaystyle \sum_{k=0}^{m-1}k^n=\frac{1}{1+n}\sum_{k=0}^n\binom{n+1}{k}B_km^ {n+1-k}$

how do we show that the values $\displaystyle \{B_n\}$ are uniquely defined? That is, in the coefficients of $\displaystyle m^{n+1}, m^n, m^{n-1}\cdots$, $\displaystyle \frac{B_0}{1+n}, B_1, B_2 \frac{n}{2}, \cdots $ the values $\displaystyle B_0, B_1, B_2, \cdots$ will always be the same regardless of the power $\displaystyle n$.