Triangle inequality raised to the power of $n$

Let $r\in [1,\infty)$ and $X$ be a random variable such that $E[X^r]<\infty$. Let $\{a_n\}_n$ be a sequence of positive real numbers such that $\lim_n a_n = 0$. My question is whether or not we have that$$\lim_{n\rightarrow\infty}E[(X+a_n)^r] = E[X^r]$$

My intuition is that it's true, but neither Fatou's Lemma or the dominated convergence theorem work.

Thanks in advance.