In the framework of time series analysis
Why does $\lim_{n \rightarrow \infty} n^{-1} \sum_{|h| <n} |\gamma(h)| = \lim_{n \rightarrow \infty} 2|\gamma(n)| $?
The RHSLHS (left hand side) sequence of functions we are taking the limit of is $$\{ \gamma(1), \frac{1}{2}(\gamma(1) + \gamma(2)), \frac{1}{3}(\gamma(1) + \gamma(2) + \gamma(3)), \dots \} $$$$\{ |\gamma(1)|, \frac{1}{2}(|\gamma(1)| + |\gamma(2)|), \frac{1}{3}(|\gamma(1)| + |\gamma(2)| + |\gamma(3)|), \dots \} $$
The LHSRHS (right hand side) sequence we are taking the limit of is $$\{2 \gamma(1), 2 \gamma(2) , 2 \gamma(3), \dots \}$$$$\{2 |\gamma(1)|, 2 |\gamma(2)| , 2 |\gamma(3)|, \dots \}$$
Adding some steps in-between might help me greatly.
Where $\gamma(h)$ is the auto-covariance function defines as $\gamma(h) \equiv Cov(X_{t+h}, X_t)$.