Martingale and quadratic variation inequality

Question

I have the following inequality $$\mathbb{E}(\mid[M^{\Pi^m},M^{\Pi^m}]_T^{1/2}-[M^{\Pi^n},M^{\Pi^n}]_T^{1/2}\mid^p)\leq \mathbb{E}([M^{\Pi^m}-M^{\Pi^n},M^{\Pi^m}-M^{\Pi^n}]_T^{p/2}),$$ where $M$ is a martingale and $\Pi^n$ gives a partition of $[0,T]$ and $p\geq 1$. I can't see why this is true for all $p\geq1$.