Background:

The weak law of large numbers states that for a sequence $X_1,X_2,\ldots,X_n$ of iid RVs, with expectation $\mu$ and variance $\sigma^2$, the sample mean converges to $\mu$:

$$\hat{X}=\frac{\sum_{i=1}^nX_i}{n}\stackrel{p}{\rightarrow}\mu$$

That is the sample mean converges in probability to the population mean as the number of RVs approaches $\infty$.

Question:

How can you obtain infinitely many iid random variables from a finite population? How do you in practise check that your random variables are independent?

Edit:

By finite population I mean that you consider a population of individuals. This population is finite. You consider a characteristic in the population. You model the characteristic with a random variable. I do not mean that the range of the random variable is finite.