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