E^2={x^2:x∈E}.Prove that E^2 is measurable whenever E is a measurable subset of R.

Question

I have already proven that if Φ(x) is a strictly increasing Lipschitz function then Φ(E) is measurable whenever E is a measurable subset of R. And then by use it I proved that $\sqrt{E}$ is measurable whenever E is a measurable subset of R. So maybe I should construct a Lipschitz function based on $x^{2}$, but how ? And how about $E^{3}$