 Solution

With one true regressor and one additional candidate regressor, the reduction in residual sum of squares from fitting the larger nested model is $Z^2\sim {\chi }_1^2$. AIC chooses the wrong larger model exactly whenwhose probability is fixed and positive, independent of $n$. BIC chooses it whenFor $n>e^2$ this probability is strictly smaller than the AIC error probability, and it tends to zero as $n\to \infty$.