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

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