# Sequence of random variables and convergence in probability.

Let $$\Omega:=[0,1]$$ with Lebesgue measure.Let's define a sequence $$X_n= \frac{\omega}{n}$$. Check:

a) Convergence in distribution

b) Convergence in probability

My solution: b) $$P(|X_n-X| \ge \epsilon)=P(|X_n| \ge \epsilon)=P(\frac{\omega}{n} \ge \epsilon)=P(\omega \ge \epsilon n )$$ And this is equal 0 for each $$\epsilon >0$$ and $$n \to \infty$$.

a) $$X \equiv 0$$

We have $$F_X(t)= \begin{cases} 0 &\text{for } t < 0\\1 &\text{for } t \ge 0\end{cases}$$

Now we have to calculate $$F_{X_n}$$ and here I have problem.