This is a homework question for a course in asymptotic statistics.
Definitions.
Let $X_1, X_2, \ldots$ be sequence of random variables with $|X_n| < B$ almost surely for all $n$.
Let $X_n \xrightarrow[]{p} X$ denote convergence in probability: $\lim_{n\to \infty} P(|X_n -X| > \epsilon ) = 0.$
Let $X_n \xrightarrow[]{r} X$ denote convergence in the rth mean: $\lim_{n\to \infty} E(|X_n -X|^r) = 0.$
Let $r > 0$.
Problem.
Show that $X_n \xrightarrow[]{p} X$ if and only if $X_n \xrightarrow[]{r}X$.
Work.
I have a proof for the if part, but not for the "only if" part: $$X_n \xrightarrow[]{r} X \Leftarrow X_n \xrightarrow[]{p} X.$$
I tried to show that $E|X_n - X|^r \leq P(|X_n-X| \geq \epsilon)$.
First: I tried expanding $|X_n - X|$ using the binomial theorem. Could only find a bound in terms of a sum over $B$.
Second: Tried writing the expectation in terms of a probability density integral. Again I can only find a bound in terms of $B$.
Third: I tried splitting the $E|X_n - X|^r$ using indicator functions. I ended up getting that $E|X_n - X|^r \geq E|X_n - X|^r$. I don't see how to do this better. I feel like it should involve $B$ somehow.
Status.
I am very stuck.
