LEMMA: For all integers , the polynomial is nonnegative for all .

Proof. First note that , so is a root. Taking the derivative of this function gives . When , , so the overall slope of the function is negative. At , the slope of the function is , and when , the slope of the function is positive. Therefore, the function can never dip below since it is an absolute minimum along the positive real numbers.

Subtracting the second equation from the first gives


so

\[\begin{align*}x_2^2-2x_2+x_3^3-3x_3+2+\cdots+x_n^n-nx_n+(n-1)&=-\dfrac{n(n-1)}{2}+\left[1+2+\cdots+(n-1)\right]\\&=0.\end{align*}\]

This can be written more concisely as From the Lemma, it is known that all the individual "terms" of this summation are nonnegative. When nonnegative expressions add up to , each of them must be equal to . Since it was established in the Lemma that is a root and a relative minimum of the expression for all , we have that for . Finally, plugging our known values into the first equation gives , so as well. Hence is the only solution to the system.