It is known that from the CLT, if $X_i \stackrel{\text{iid}}{\sim} F$ for some distribution $F$ with finite variance, then $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i - \text{E}[X]) \stackrel{d}{\to} N(0,\sigma^2)$$ for some $\sigma^2$. Now, define $n$ different sequences of random variables of the form $\{A^i_k\}_{k=1}^\infty$ such that $A^i_k \stackrel{p}{\to} 1$ as $k\to\infty$ for all $i=1,2,\ldots,n$.
Here is my question. Is the following statement true? I really want to apply Slutsky's theorem in some form, but I can't seem to do it because there are different variables being multiplied at each term in the sum. $$\frac{1}{\sqrt{n}} \sum_{i=1}^n (X_i A^i_n - \text{E}[X]) \stackrel{d}{\to} N(0,\sigma^2) \quad \text{as } n\to\infty$$