Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=0$, we have $x^{0}-y^{0}=0$ which is divisible by $x-y$, thus $P(0)$ is true.
3. Assume $P(k), k\gt0$ is true, that is $x^{k}-y^{k}$ is divisible by $x-y$.
4. For $n=k+1$, we have $x^{k+1}-y^{k+1}=x\cdot x^{k}-x\cdot y^{k}+x\cdot y^{k}-y\cdot y^{k}=x(x^{k}-y^{k})+y^{k}(x-y)$
which is divisible by $x-y$.
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.