Prove $x_n$ converges if $x_n$ is a real sequence and $s_n=frac{x_0+x_1+cdots+x_n}{n+1}$ converges

Given that $x_n$ is a real sequence, $s_n = \frac{x_0+x_1+\cdots+x_n}{n+1}$ and $s_n$ converges, $a_n = x_n-x_{n-1}$, $na_n$ converges to 0, and $x_n-s_n=\frac{1}{n+1}\sum_{i=1}^n ia_i$, prove $x_n$ converges, and that $x_n$ and $s_n$ converge to the same limit.

Since I don't know anything about the sequences increasing, decreasing, or being positive, I was thinking that I can show that $\frac{1}{n+1}\sum_{i=1}^n ia_i$ converges, so then $x_n$ will be the sum of two convergent sequences and so it must converge, but I don't know how to show $\frac{1}{n+1}\sum_{i=1}^n ia_i$ converges, or if this is even the right way to go about this problem.

Also since $s_n$ converges, it is bounded so maybe I can use that fact, but I'm not sure how that could help.

I also showed that $a_n$ must converge to 0 as well using the squeeze theorem, but I don't know if that helps either.

Any hints would be great. I have been stuck on this for days.