Weak law of large numbers in finite populations [closed]

Question

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.

Edit 2

We know that $\mu$ is a population characteristic. Let us assume the population is of size $n$. Denote by $Y$ the random variable that describes the population characteristic. Then $\mu=\text{E}(Y)$. Let $Y_1,\ldots,Y_n$ denote the random variables of respectively individual 1 to $n$. By definition $\mu=\frac{\sum_{i=1}^nY_i}{n}$. We then make a sample from the population. How can we obtain a sample of size $n+1$ or $n\rightarrow\infty$ that is iid from a population that is finite? Here some say that we can sample from $Y_1,\ldots,Y_n$ WITH replacement.

Edit 3

If we consider a sample of size $N$ with $N\leq n$, where the sampling is done without replacement, can we can obtain a iid sample if the original RVs are independent? If $Y_1,\ldots\ Y_n$ are iid and we let $X_1=Y_1,\ldots, X_n=Y_n$ and consider any subset of $X_1,\ldots, X_n$, then this subset will consist of iid RVs, right? Am I missing some point here?

Answer 1

So if your question is "how can we apply the LLN with finite data" then I give the answer I gave in the comments:

"So you don't ever have infinitely many. It's that after you have a large finite number, you get `close' to the limit value of $\mu$"

But, if your question is rather: How do we reconcile the LLN with a finite population, then I lead to to this fantastic post about finite population sampling :Are "random sample" and "iid random variable" synonyms?

The other post about $n\to N$ being equivalent to $n\to\infty$ is problematic as LLN is a result of probability limits and sequences and those necessarily are not finite in index.

So n is only ever able to reach N if we sample without replacement, meaning the samples are no longer iid. If we sample with replacement then we never run out of n.