Let $X$ be a random variable such that there exists a $r \in (0, \infty)$ where $E(|X|^r)<\infty$. Show that $E(|X|^l)<\infty$ for all $l \in (0, \infty)$ and $l \le r$.
My attempt is like this: Fix $l \in (0, r]$. Then $|x|^l \le |x|^r$ for all $x$. So taking expectations, we have $E(|X|^l) \le E(|X|^r) < \infty$, and we are done.
However, I was given a hint where $|x|^l < |x|^r+1$ and then taking expectations yields $E(|X|^l) < E(|X|^r)+1 < \infty$. Why do we need to have a $+1$ here? Is my approach fine as well?