discrete mathematics - Proof: For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$

I need help proving the following statement:

For all integers $x$ and $y$, if $x^3+x = y^3+y$ then $x = y$

The statement is true, I just need to know the thought process, or a lead in the right direction. I think I might have to use a contradiction, but I don't know where to begin.

Any help would be much appreciated.

Answer

We have
$$\begin{eqnarray*} x^3+x=y^3+y&\Longleftrightarrow& (x^3-y^3)+(x-y)=0\\ &\Longleftrightarrow& (x-y)(x^2+y^2+xy+1)=0. \end{eqnarray*}$$
Since $x^2+y^2+xy+1=(x+\frac{y}{2})^2+\frac{3}{4}y^2+1>0$, we get $x=y$.
The hypothesis $x,y$ are integer numbers is redundant.