By subtraction
Now take your limits
...But I can't get it
We have a sequence of iid nonnegative rv's
Assuming that is finite, we have by the sLLN that converges to a finite limit.
But how can I conclude that ?
I tried to relate it to the series , but the inequality is the other way round than the one that would be helpful...
...so since inequalities don't help me, I hope you guys can help me
Thanks !
Note : I also - miserably - tried Cesàro's mean.
Look at Theorem 2 on page 125 of..........
Probability theory: independence ... - Google Books
MathEagle's proof works fine : almost surely,
- the sequence converges to
- the sequence converges to (because of the first point)
- the sequence converges to 1 (obviously),
and these three events together (I could also have let the third one appart) imply that converges to 0, hence this latter limit holds almost-surely.
By the way, I'm not sure how to use Borel-Cantelli here.