1. ## need help

hi all i need the proof of the following question but i dont know from where i get it please help me
1. if xn converge to x uniformly then xn converge to x in r-th mean
2. if xn converge to x almost sure then xn converge to x in probability
3.if xn converge to x every where then xn converge to x almost sur

2. Originally Posted by starmh
hi all i need the proof of the following question but i dont know from where i get it please help me
1. if xn converge to x uniformly then xn converge to x in r-th mean
2. if xn converge to x almost sure then xn converge to x in probability
3.if xn converge to x every where then xn converge to x almost sur
1. I believe that reverse Fatou's lemma should do the trick.

2. Consider the event $\{ |X_n-X|>\epsilon \mbox{ i.o.}\}$ which is a superset of $\{|X_n-X|>\epsilon\}$. Show that this set is disjoint from the set $\{ \omega: \lim X_n(\omega)=X(\omega)\}$.

3. What is this set equal to $\{\omega: \lim X_n(\omega) \neq X(\omega)\}$ ? When you figure it out, the measure of it will become obvious.

3. Most books cover the various modes of convergence...
Probability theory: independence ... - Google Books