Problem
Prove that $X_n \sim N(0,\frac{1}{n})$ converges in probability to 0.
Discussion
There a problems similar to this question, but I want to ask specifically about the convergence of an integral that arises in the solution.
Attempt
Let $\epsilon>0$. To show $X_n \overset{p}{\to} 0$, we must show $P(|X_n| > \epsilon) \to 0$.
$$ \begin{align} P(|X_n|>\epsilon) &= P(X_n>\epsilon) + P(X_n < -\epsilon)\\ &= P(Z>\epsilon\sqrt{n}) + P(X_n<-\epsilon\sqrt{n}) \end{align} $$ where $Z$ is the standard normal distribution. Thus we want to show $$P(|X_n|>\epsilon)=\int_{\epsilon\sqrt{n}}^\infty\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\text{dx} + \int_{-\infty}^{-\epsilon\sqrt{n}}\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}\text{dx}\to 0$$
I understanding intuitively that these integrals should converge to zero, but how can I show rigorously that they go to zero?