Show that the probability of a limit goes to one (1)

Question

This seems a fairly trivial exercise, but I want to see how to show this intuitive result.

We are given $Y\in N(0,2-2\rho)$, where $\rho$ is a number (actually a correlation coefficient). I want to show that for any $\epsilon>0$

$$P(|Y|\leq\epsilon)\rightarrow1$$

if $\rho \uparrow 1$.

Intuitively this result feels like it should be true, since it makes sense: In the limit the variance approaches zero (0), so all probability mass gets concentrated on the mean $\mu=0$. Do I need to apply some kind of convergence mode for this?

Answer 1

\begin{align} P((|Y| \le \epsilon ) &= P((|Y| \le \frac{\epsilon}{2-2\rho}\cdot (2-2\rho) ) \\ &\ge 1- \frac{1}{\left(\frac{\epsilon}{2-2\rho} \right)^2} \\ &=1-\left(\frac{2-2\rho}{\epsilon} \right)^2 \end{align}

Now, just let $\rho$ tend to $1$.