David Williams "Probability with Martingales" : Proof of Monotonocity of $L^p$ norms

Question

I find the author often leaves out 'obvious' steps..

How to fill in the missing steps in the proof below ?

I can see that $X_n \uparrow {|Y|}^{p}$ so through Monotone convergence $E(X_n) \uparrow E(|Y|^p)$, but what then ? is it that $E(X_n) \le {E(X_n)} ^ {r \over p}$ since both sides positive and r > p ?

Answer 1

You have $$ E(X_n)^{1/p} \le ||Y||_r$$ and want to show $$||Y||_p\le ||Y||_r. $$ $X_n$ is a monontone sequence of RVs converging to $Y^p$ so by MON, $E(X_n)\to E(Y^p)$ which means $$ E(X_n)^{1/p}\to E(Y^p)^{1/p}=||Y||_p$$ so by the first inequality, $$ ||Y||_p=||Y||_r$$