TITLE: Weak Law of Large Numbers for Dependent Random Variables with Bounded Covariance
QUESTION [36 upvotes]: I'm currently stuck on the following problem which involves proving the weak law of large numbers for a sequence of dependent but identically distributed random variables. Here's the full statement:

Let $(X_n)$ be a sequence of dependent identically distributed random variables with finite variance. 
Let $\displaystyle S_n = \sum_{i=1}^n X_i $ denote the  $n^\text{th}$ partial sum of the random variables $(X_n)$.
Assume that Cov$(X_i,X_j) \leq c^{|i-j|}$ for $i, j \leq n$ where $|c| \leq 1$. 

Is it possible to show that $\displaystyle \frac{S_n}{n} \rightarrow \mathbb{E}[X_1]$ in probability? In other words, is it true that given any $\epsilon>0$,
$$ \lim_{n\rightarrow \infty} \mathbb{P}\bigg[\Big|\frac{S_n}{n} - \mathbb{E}[X_1]\Big| > \epsilon\bigg] = 0$$
EDIT: Following some comments, it turns out that I had the right approach so I've gone ahead and answered my own question below.

REPLY [4 votes]: Sorry again to unearth the topic, but I just can't leave a comment.
Afaik, the original result was published in S. Bernshtein, “Sur la loi des grands nombres”, Communications de la Société mathématique de Kharkow. 2-ée série, 16:1-2 (1918), 82–87. In hard-to-read form, as befits a classic.
Short proof (similar to proposed) of initial statement and generalizations for Riesz Means are in paper of V. V. Kozlov, T. Madsen, A. A. Sorokin, “On weighted mean values of weakly dependent random variables”, Moscow Univ. Math. Bull., 59:5 (2004), 36–39. Moreover, the note that conditions:
(1). $V[X_i]≤c$
(2) $\lim\limits_{|i-j| \to +\infty} \mathrm{Cov}(X_i, X_j) = 0$
can be weakened to:
(1'). $\sum\limits_{i=1}^n\mathbf{V}[X_i] = o(n^2)$
(2'). $|\mathrm{Cov}(X_i, X_j)| \le \varphi(|i-j|)$, where $\frac1n\sum\limits_{i=1}^n\varphi(i) \to 0$.
Unfortunately, both of this links are in Russian.