For a time series, does finite variance imply finite autocovariance?

Question

If variance is finite at all times, does this imply that all pairwise autocovariances are also finite?

Answer 1

For any times $s$ and $t$, the Cauchy-Schwarz inequality gives us $$ |\text{Cov}(X_s,X_t)|^2 \leq \text{Var}(X_s)\text{Var}(X_t) < \infty. $$

If you're worried about the means being possibly infinite/undefined, the discussion here might be helpful.