I keep seeing sources stating, without proof, that the standard deviation of the sampling distribution of the sample mean:
$$\sigma/\sqrt{n}$$
is an approximate formula that only holds if the population size is at least 20 times the sample size.
Can anyone offer a proof of this statement, or disprove it? If it's false, could you please explain the intuition behind why anyone would come up with this in the first place?
Here are two places in which I've seen this claim:
1) "You often see this "approximate" formula in introductory statistics texts. As a general rule, it is safe to use the approximate formula when the sample size is no bigger than 1/20 of the population size." - https://stattrek.com/sampling/sampling-distribution.aspx
2) "the formula for the standard deviation of the sampling distribution of the sample mean, $\sigma/\sqrt{n}$, holds approximately if the population is finite and much larger than (say, at least 20 times) the size of the sample". A business statistics textbook.
EDIT:
Yes, this is in the context of sampling from finite populations.