Assume that $X_1, X_2, \dots$ are independent real random variables with means $\mu_1, \mu_2, \dots$ and variances $\sigma_1^2, \sigma_2^2, \dots$ such that $\displaystyle\sum_{k=1}^\infty \dfrac{\sigma_k^2}{k^2} < \infty$. Let $\overline{X_n}=\frac{1}{n}\sum_{k=0}^n X_k$ and assume that $\mu_n\rightarrow \mu$.
Show that $E[\sup_n |\overline{X_n} - \mu|]<\infty$.
My attempt:
From Kolmogorov's SLLN, we know that $\overline{X_n}\rightarrow \mu$ almost surely. It seems important, but I don't see how to use it...
I also thought that I might be able to get there by using the general inequality $E[|X|]\leq \sum_{k=0}^{\infty} P(|X|>k)$ and Chebyshev's inequality $P(|X|>k)\leq \frac{\sigma^2}{k^2}$. However, relating $\sup_n |\overline{X_n} - \mu|$ to the variance of just one of the $X_k$ is difficult.
Any help would be appreciated!