Let $ X_{1}, \cdots, X_{n} $ be a sequence of dependent random variables having the same finite mean $ \mu=\mathrm{E}\left(X_{1}\right), $ the same finite variance $ \sigma^{2}=\operatorname{Var}\left(X_{1}\right), $ and $ \left|\operatorname{Cov}\left(X_{i}, X_{j}\right)\right| \leq \frac{1}{(i-j)^{4}} $ for $ i \neq j . $ Show that $ \bar{X} $ converges in probability to $ \mu . $
I am stuck on the upper bound of the sum of the covariances: $\sum_{i=1}^{n} \sum_{j \neq i,j=1}^{n}|{\rm Cov}(X_i,X_j)|\le2 \cdot \sum_{k=1}^{n-1}(\frac{n-k}{k^4}) $
How to show that $n^2$ is a proper upper bound of the sum?
Since $\left\lvert\operatorname{Cov}(X_i,X_j)\right\rvert\leq (i-j)^{-4}\leq 1$, summing over all $n(n-1)$ ordered pairs $(i,j)$ with $i\neq j$ gives the easy upper bound $n(n-1)<n^2$.