# Role of random sample assumption in consistency of OLS estimator

I guess in part what this all amounts to is what does the assumption {(x_i,y_i) : i=1,2,...,n} being i.i.d. imply about the i.i.d-ness of functions of it?

I am confused because for example I have seen online that the product of i.i.d. random variables are not necessarily i.i.d.

Thanks for all the help!

TLDR: $\{X_iY_i\}$ is i.i.d. as well, so the standard argument for consistency carries out here.
From the premises of the question, we know that $\{(X_i,Y_i)\}$ is iid.
1. Independence of $X_iY_i$ from $X_jY_j$, $\forall i\ne j$. We can apply a standard result in probability theory that states that (measurable) functions of independent variables are independent as well (here's a concise proof of this result).
2. Identical distribution. It follows from the definition of the distribution function of a function of a random vector ($(X_i,Y_i)$ in this case) and the fact that $(X_i,Y_i)\overset{d}{=}(X_j,Y_j)$ for all $i,j$.