The vague statement that "sampling with and without replacement from a finite population are approximately equivalent when the sampling fraction is small" is given a precise meaning in terms of limit theorems for distributions in $R^\infty$ and $D\lbrack 0, \infty)$.
"A Note on the Asymptotic Equivalence of Sampling with and Without Replacement." Ann. Statist. 2 (4) 819 - 821, July, 1974. https://doi.org/10.1214/aos/1176342770