Simple problem with convergence in distribution.

Question

Let ${\displaystyle X_{n}\ {\xrightarrow {\overset {}{d}}}\ a\quad}$, where a is constant.

Is that true that $P(X_n<a) \rightarrow 0$ ?

My intuition tell's me that this is true so i tried to prove it

I tried to do something like this :

$P(X_n<a)=1-P(X_n=a)-P(X_n>a)$

And I stuck at this moment.

Can you please help me with this ?

Answer 1

Let $X_n\sim N(0,1/n)$. Then $X_n\stackrel{d}\to 0$, but $P(X_n<0)=\frac12\not\to 0$.