How to show that the estimator is consistent? $Y_i=\beta_0+\beta_1X_i+U_i$ is my regression model for an I.I.D. sample with N=1000 observations. Suppose $U_i\sim I.I.D.(0,\sigma^2)$ and Xi are also I.I.D for i=1,2,3......1000. Xi is independent of Ui. How to show that the estimator $\beta_1={{Y_3-Y_2}\over {X_3-X_2}}$ is a consistent estimator of the OLS estimator?
Simplication gives 
$\tilde {\beta}={{\beta_0+\beta_1X_3+U_3-\beta_0-\beta_1X_2-U_2}\over {X_3-X_2}}={{\beta_1(X_3-X_2)+U_3-U_2}\over {X_3-X_2}}=\beta_1+{{U_3-U_2}\over {X_3-X_2}}$
What is the next step?
Remember that consistency describes how the estimator behaves in the limit as N asymptotically approaches infinity. Assuming no errors in your math up to this point, you need to consider how your error terms $U_i$ behave asymptotically as well.