This is part of a convergence theorem for $L_p$ martingales.
Theorem. We have $p > 1$ and $B>0$. Let further $(X_n)_n$ be a martingale with $\sup_{n \in \mathbb{N}} E[|X_n|^p] \leq B < \infty$. Then there exists a random variable $X_{\infty}$ such that $E[|X_{\infty}|^p] \leq B$ and s.t.
$X_n \longrightarrow X_{\infty}$ a.s. and in $L_p$ norm.
A powerful theorem but with a strong requirement condition. My question concerns this requirement. Why do we demand that
$\sup_{n \in \mathbb{N}} E[|X_n|^p] \leq B < \infty$ $(*)$
instead of just
$\sup_{n \in \mathbb{N}} E[|X_n|^p] < \infty$ $(**)$
I know that, in general, these are different concepts. For example, only demanding that for all $n \in \mathbb{N}$
$|X_n| < \infty$ does not imply that $\sup_{n \in \mathbb{N}} |X_n| < \infty$
But in the case of $(*)$, we already have a supremum. And in my understanding, if $(**)$ holds, we must have some $B>0$ such that $(*)$ must also hold. Since if the supremum does not go to infinity, it must be smaller than some value.
Where am I going wrong?
$(*)$ and $(**)$ are both equivalent in that they tell you the sup is finite. $(*) $ gives a name to the sup ($B$), which can be used later.
So for a sequence $(a_n)$ both conditions: $\sup_n a_n<+\infty$ and $\sup_n a_n\le B<+\infty$ are equivalent. The latter obviously is equivalent to: $a_n\le B$ for all $n$, which is of course strictly stronger than: $a_n<+\infty$ for all $n$.