Use the triangle inequality for the norm on $L^{q}$ to show that if $q \geq 1$ and $X_{n} \rightarrow^{q.m} X$ (converges in q mean) then $$E[|X_{n}|^{q}] \rightarrow E[|X|^{q}]$$
So far, I've tried figuring out how to extend the triangle inequality to powers of $q$. I've tried relating $$|X_{n} - X|^{q}$$ to different values, but I'm not sure if the inequality $$|X_{n} - X|^{q} \leq |X_{n}|^{q} - |X|^{q}$$ holds in all scenarios. I was trying this because I'm assuming that once I can get a relation between different expressions, it is simple to extend this to expectation. I've also tried experimenting first with the simple triangle inequality by noting that $|(X_{n} - X) + X| \leq |X_{n} - X| + |X|$ but again can't extend this to powers of q. I'm not entirely sure where to start exactly.