I'm studying for a test on Monday and am going through some supplementary problems. These problems do not come with solutions provided but I still think they are very good practice.
Suppose sequence $X_n$ satisfies:
\begin{align} \lim_{n} \mathbb{E}[|X_n - c|^\alpha ] = 0 \end{align} where $\alpha > 0$. Show that the sequence converges to $c$ in probability.
This is a type of problem I have never seen before. I am sort-of familiar with convergence in probability problems but I have never seen one that takes the limit of an expectation.
I know for $X_n$ to converge in probability to some value $a$ we must have: \begin{align} \lim_{n} \mathbb{P}(|X_n - a| > \epsilon) = 0 \end{align}
and this looks very similar, but the expectation is still throwing me off.
Any tips on how to begin? I will try my best to understand but again these problems were not assigned at any time during my course so far and so I am not even sure if the proper ways to deal with this problem were covered in class.
Also, why is it important to note that $\alpha$ is positive?