Let {$X_i;i\geq0$} be a sequence of random variables defined on the probability space ($\Omega,\mathcal{F},P$). If $||.||_p$ is the $L^p$ norm defined as $||X_i||_p=(E[|X_i|^p])^{1/p}$, how should I prove that $||X_i||_p\leq||X_i||_2$ for $1\leq p\leq2$?
Does it follow from Jensen's inequality?
You are right. It follows from Jensen's inequality. By Jensen's inequality, if $Z$ is a random variable and $\varphi$ is a convex function, then $$ \varphi\left(\operatorname EZ\right) \le \operatorname E\varphi(Z). $$ Let $\varphi(x)=x^{2/p}$ with $p\le 2$ and $Z=|X|^p$. Then $$ (E|X|^p)^{2/p}\le E|X|^2. $$ Taking the square root of each side gives the result.
Alternatively, the same inequality can be obtained using Hölder's inequality.