# Convergence of ratio of population mean and sample of sample means to a fixed constant

Let $A = \lbrace a_1, ..., a_n \rbrace$ be a set of numbers in $\Bbb R$. Let $\omega = \lbrace x_i \rbrace_{i=1}^k$ be a random sample of $k$ distinct elements in $A$. Let $\bar{\omega} = \frac{\sum_{i=1}^k x_k}{k}$. Suppose we aggregate $b$ random samples from $A$ each with $k$ observations and compute the corresponding $\bar{\omega}$ for each. Then we would have a set $\Omega = \lbrace \bar{\omega}_1, ...,\bar{\omega}_b \rbrace$.

I have a data set. Through experimentation, I realized

$\displaystyle\lim_{b\to \infty}\frac{\text{Var}(A)}{\text{Var}(\Omega)} \rightarrow k$

I assume this follows from the central limit theorm. Why is this the case?