## Probability convergence with OLS estimators - univariate vs bivariate

Hey guys, this is my first post here. I was hoping you could help point out how I should go about tackling this proof:

Let ( $Y_1$, $X_1$, $_1$, $X_1$, $_2$)...,( $Y_n$, $X_n$, $_1$, $X_n$, $_2$) be an i.i.d. sample from ( $Y$, $X_1$, $X_2$) satisfying:

$Y$ = $\beta_0$ + $\beta_1$ $X_1$ + $\beta_2$ $X_2$ + $U$

where (1, $X_1$, $X_2$) is not perfectly collinear, $E[Y^4]$ < $\infty$ and $E[X^4]$ < $\infty$ (** - please note it should be $X_j$ in the above, not just $X$, but my Latex-fu is weak).

Suppose a professor estimates the equation: $Y$ = $\beta_0$* + $\beta_1$* $X_1$ + $U$*

by OLS. Show that:

$\hat\beta_1$* p $\to$ $\beta_1$ + $\beta_2$ Cov[ $X_1$ , $X_2$]/Var[ $X_1$].

Also, under what conditions will it be true that $\hat\beta_1$* is consistent for $\beta_1$? (Here I'm confused - it's obvious that (X'X)^-1 is well defined since they already explicitly stated that (1, $X_1$, $X_2$) is not perfectly collinear and that $E[Y^4]$ < $\infty$ and $E[X^4]$ < $\infty$).

Any advice to get me started?