$E[|X_n|] < \infty$ $\forall n$ vs $\sup_{n \ge 0} E[|X_n|] < \infty$

Question

I understand that just because a sequence of random variables are integrable that does necessarily imply that their expected value is a.s finite. Take for example, $E[|X_n|] = n$ $\forall n$. However, my question is that if we given somewhat a stronger condition: $\sup_{n \ge 0} E[|X_n|] < \infty$ can we conclude that $\{E[|X_n|]: n \ge 0\}$ is a bounded sequence? In other words, is the statement $ \sup_{n \ge 0} E[|X_n|] < \infty \Leftrightarrow E[|X_n|] < K$ $\forall n$ and for some $K \in \mathbb{R}$.

Answer 1

Yes. There's in fact no probabilistic content there, and it may be more clear if we let $a_n = E[|X_n|]$. For any sequence of real numbers, $\sup_{n \geq 0} a_n < \infty$ is equivalent to $a_n < K$ for all $n$ for some $K \in \mathbb{R}$.