$(X_n)$ is a sequence of $L^2$ random variables with $EX_n=0$ for all $n$ and suppose there is a constant $c$ s.t. $\operatorname{Var}(X_{n+k}−X_n)\leq ck$, for all $n,k\geq0$. Show that $X_n/n$ converges to $0$ a.s. (Hint: First prove along a suitable subsequence).
I can see that we are trying to make the probabilities summable along a subsequence but may I know how to choose a subsequence so that the upper bound we use for variance makes the probabilities summable?
$$P(|X_{n_k}|>nϵ)≤\operatorname{Var}(X_{n_k})/{n_k}^2ϵ^2\leq?$$
from Hot Weekly Questions - Mathematics Stack Exchange
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