convergence exercise in law (stuck)

Question

I have this exercise

Let $(Xn)$ be a sequence of r.v whose distribution is defined for $n \in N^*$ by

$P\left(X_n=1-\frac{1}{n}\right)= P\left(X_n= 1+\frac{1}{n}\right) = \frac{1}{2}$

Study its convergence in law.

My attempt :

is to show that its converge in probability (and Convergence in probability implies convergence in) so it converge in law

$P(|X_n-1|<\epsilon) = P(-\epsilon < X_n-1< \epsilon)= P(1-\epsilon<X_n<1+\epsilon)$

Is this the right start? What to do next because I'm stuck??

Answer 1

Almost surely, $X_n=1\pm \frac 1 n$ so $|X_n-1|=\frac 1 n <\epsilon$ whenever $n >\frac 1 {\epsilon}$. So $X_n \to 1$ in probability.