I have learnt the following definition of the law of large numbers:
Theorem 17.4 (Law of Large Numbers). Let $X_1, X_2, \ldots$, be a sequence of independent and identically distributed random variables with common finite expectation $\mathbb{E}\left[X_i\right]=\mu$ for all $i$. Then, their partial sums $S_n=X_1+X_2+\cdots+X_n$ satisfy $$ \mathbb{P}\left[\left|\frac{1}{n} S_n-\mu\right|<\varepsilon\right] \rightarrow 1 \quad \text { as } n \rightarrow \infty, $$ for every $\varepsilon>0$.
I am having trouble connecting this to the other common definition of the law of large numbers, which is that "the average of the results of independent identical trials approaches the trial's expected value the more trials we take", for the following single reason:
the theorem seems to state that the sum of $n$ random variables divided by $n$ approaches the expected value of one of them, whereas the other definition states that the average of any specific values which said random variables assume approaches the expected value of these random variables. I am seeing a difference between these because one talks about random variables, the other talks about a set of possible values which they assume. I would be very appreciative if anyone could explain why these definitions are the same.