"For $p > 0$" meaning for all $p$ or just one $p$?

Question

This sentence:

Let $(X_n)$ be a sequence of random variables. Let $X$ be some other variable. Let $p>0$. Show that if $\sum _{n=1} ^\infty E |X_n-X|^p$ is finite, then $X_n \stackrel{a.s} \to X$.

Is the requirement that it holds for every $p > 0$ for there to be a.s. convergence, or just one $p > 0$?

I don't need help to solve the problem, just to understand what the theorem it wants me to prove is actually saying.

Answer 1

It means that the statement is true for all $p>0$.

Answer 2

Just one $p>0$, not all $p>0$. Simple reason is that all $p$th order moments may not even exist! If you require a solution to this problem, comment.