How to prove the monotonicity of semi-norm?

Question

In the book "Malliavin Calculus and related topics", the author states that $||F||_{k,p}=(E(|F|^p)+\sum_{n=1}^kE(||D^n F||^p_{H^k})^{\frac{1}{p}}$ has monotonicity property, i.e. $||F||_{k,p}\leq ||F||_{j,q}$ when $k\leq j$ and $p\leq q$. How to prove it in general? I try to use Holder inequality to prove it but I fail to do so? Could someone provide a detailed derivation for it?