Proof of Equation by Well Ordering Principle

Suppose that the statement is not true.

Then the set $A=\left\{ n\in\mathbb{N}\mid\sum_{i=0}^{n}i^{3}\neq\left(\frac{n\left(n+1\right)}{2}\right)^{2}\right\} $ is not empty.

Since $\mathbb{N}$ is well-ordered set $A$ has a minimal element $m$.

That means that $\sum_{i=0}^{n}i^{3}=\left(\frac{n\left(n+1\right)}{2}\right)^{2}$ is true for $n<m$ and is not true for $n=m$.

From this you can deduce a contradiction.

(Start with $\sum_{i=0}^{m-1}i^{3}=\left(\frac{\left(m-1\right)m}{2}\right)^{2}$ and prove on base of that $\sum_{i=0}^{m}i^{3}=\sum_{i=0}^{m-1}i^{3}+m^{3}=\left(\frac{m\left(m+1\right)}{2}\right)^{2}$)

The conclusion is then that $A=\emptyset$ wich is exactly the statement to be proven.