I'm fond of the following false proof of the Strong Law of Large Numbers.  Let $X$ be a random variable with expected value $\mu$ and  variance $\sigma^2$, and let $X_1, X_2, \dots$ be i.i.d. copies of $X$.  Then $$\operatorname{Var} \left( \frac{1}{n} \sum_{i=1}^n X_i \right) = \frac{1}{n^2} \cdot n \sigma^2 = \frac{\sigma^2}{n} \rightarrow 0 \textrm{ as } n\rightarrow\infty $$
and since a random variable with variance 0 takes on a single value with probability 1, we must have $$\lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=1}^n X_i = \mu \textrm{ almost surely.}$$
(It's a memorable heuristic reason to tell undergraduate probability students, even if not a true argument.)