[Proof] Since $x^{2} y \geqslant \min \left\{x^{3}, y^{3}\right\}$, we have
$$x^{3}+y^{3}-x^{2} y \leqslant \max \left\{x^{3}, y^{3}\right\} \leqslant 1$$

Similarly, we get $\quad y^{3}+z^{3}-y^{2} z \leqslant \max \left\{y^{3}, z^{3}\right\} \leqslant 1$,
$$z^{3}+x^{3}-z^{2} x \leqslant \max \left\{z^{3}, x^{3}\right\} \leqslant 1$$

Thus, $\quad 2\left(x^{3}+y^{3}+z^{3}\right)-\left(x^{2} y+y^{2} z+z^{2} x\right) \leqslant 3$.