I was able to come up with this 'proof' of the strong law of large numbers for Bernoulli random variables, but it seems far too simple and short compared standard proofs and haven't been able to find any sources which follow this sort of reasoning, so I'm very doubtful. However, I can't seem to find any mistakes in my reasoning, so I'd appreciate if someone could point out any.
Let $\{ X_n \}_{n = 1}^\infty$ be a sequence of i.i.d. Bernoulli random variables with parameter $p$. Let $Y$ be the random variable $$Y = \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n X_i$$ Since $\left| \frac{1}{n} \sum_{i = 1}^n X_i \right| \le 1$, we can apply the dominated convergence theorem to get $$\textbf{E}[Y^2] = \textbf{E}\left[ \left( \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n X_i \right) \left( \lim_{n \to \infty} \frac{1}{n} \sum_{i = 1}^n X_i \right) \right] = \textbf{E}\left[ \lim_{n \to \infty} \frac{1}{n^2} \sum_{i = 1}^n \sum_{j = 1}^n X_i X_j \right] = \lim_{n \to \infty} \textbf{E}\left[ \frac{1}{n^2} \sum_{i = 1}^n \sum_{j = 1}^n X_i X_j \right] $$ Moving expectation into the finite sums, $$\textbf{E}[Y^2] = \lim_{n \to \infty} \frac{1}{n^2} \sum_{i = 1}^n \sum_{j = 1}^n \textbf{E}[X_i X_j] = \lim_{n \to \infty} \frac{1}{n^2} \sum_{i = 1}^n \sum_{j = 1, j \ne i}^n p^2 = \lim_{n \to \infty} \frac{(n^2 - n)p^2}{n^2} = p^2$$ The terms in the double sum where $i = j$ which would become $p$ rather than $p^2$ can be ignored since there are only $n$ of them and so are dominated by the $1/n^2$. After showing that $\textbf{E}[Y] = p$, $$\textbf{Var}[Y] = \textbf{E}[Y^2] - \textbf{E}[Y]^2 = p^2 - p^2 = 0$$ which implies that $Y$ is constant almost surely. Since $\textbf{E}[Y] = p$, we have $Y = p$ almost surely so that $$\frac{1}{n} \sum_{i = 1}^n X_i \overset{a.s.}{\longrightarrow} p \text{ as $n \to \infty$}$$ Which is the strong law of large numbers.
The only step I can think of which is possibly faulty is defining $Y$ as the limit of random variables, without knowing if this actually exists. If this is the case, I feel we could still use $\liminf$ or $\limsup$ and apply Fatou's lemma to get a two sided inequality with $p$.