Why this sequence of R.V converge in distribution, but doesnt in probability Why this sequence of R.V converge in distribution, but doesnt in probability?
Probability space $([0,1],B,m)$. ($B$ consists of all Borel sets of $[0,1]$, $m$ is the Lebesgue measure.)
Let $X_{2n}(ω)=ω$, $X_{2n−1}(ω)=1−ω$
My intuiton says that I have to show that $X_{2n}$ and $X_{2n−1}$ have the same distribution, which is an Uniform distribution. And after this, showing that they converge in distribution, which is obvious.
But How can I show this?
And also, how can I Show that $X_{n}$ does not converge in probability.
Any help guys?

Let $Y(\omega)=\omega$ and $Z(\omega)=1-\omega$. Then $X_{2n}\to Y$ in probability and  $X_{2n-1}\to Z$ in probability. If $X_n$ converges in probability then we must have $Y=Z$ almost surely which is false since $P(Y=Z)=0$. 
