Let $$\overline{X_n}:=\max_{0 \leq m \leq n} X_m^+$$ for a sequence of random variables $X_i, i \geq 1$ (in fact, it is a submartingale),
where $X_m^+:=\max(X_m,0)$.
We want to show that $$||\overline{X_n}||_p \leq \frac{p}{p-1} ||X_n||_p.$$
To do so, our first step is to state that we can assume wlog that $X_n \geq 0$ for all $n$, otherwise we would just replace $X_n$ by $X_n^+$.
So what this means is that we need to show that if it holds for $X_n^+$, then it holds for $X_n$ as well, right? So why does $$||\overline{X_n^+}||_p \leq \frac{p}{p-1} ||X_n^+||_p$$ imply $$||\overline{X_n}||_p \leq \frac{p}{p-1} ||X_n||_p?$$
What I thought of:
$$||\overline{X_n}||_p = ||\overline{X_n^+}||_p \leq \frac{p}{p-1} ||X_n^+||_p.$$
But I do not get how I can conculde $$||X_n^+||_p \leq ||X_n||_p.$$
Is my idea completely wrong? Can you help me why we can assume non-negativity wlog?
Note that $$|X_n| \geq \max\{X_n,0\} = X_n^+ = |X_n^+|$$ implies $$\int |X_n|^p \, d\mathbb{P} \geq \int |X_n^+|^p \, d\mathbb{P}$$ for any $p \geq 1$. Hence, $$\|X_n\|_p \geq \|X_n^+\|_p.$$