[SOLVED] Some problems (limsup, liminf, ...)

Hi ^^

Probability again ! :D

So now we're dealing with Borel-Cantelli's lemma, limsup and liminf, and associates... And this is desperating to see how it is difficult to grasp the concept :D

We'll correct these exercises next week and we don't have to solve them before, but I'm not kind of patient (Rofl) I tried to do them today, but nothing came out from these ones... Just getting hints would really help me go through... ^^

By definition,

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

**1.**

is a sequence of iid Bernoulli(p), p (0,1), random variables

- show that there is almost everywhere an infinity of n such that

>>> So basically, I have to show that , that is to say , where

I thought I could use the inequality

But I don't see how to get the RHS go to 1... (I told you I have problems with limsup :D)

- show that for any n, if we define , there is, almost everywhere, only a finite number of that are realised.

>>> huh ???

**2.**

Let be a similar sequence as above.

Let and

Show that converges in probability to

>>> so I have to show that

and then... ? XD

thanks in advance ^^