I have problem understanding the difference when I look at the alternative definition of a.s. convergence. I know how it is defined originally, but it is the alternative definition which makes it easier to compare it to convergence in probability:
Convergence in probability is defined that for any $\epsilon >0$.:
We see there that the only difference is the sup. But even with the sup, I struggle to see the difference, can someone explain to me where I see it incorrect:?
For instance, if I look at convergence in probability, I would think like that. I choose an $\epsilon$. Then for any $\epsilon_2$, there is an N, such that if $n \ge N$, then $Pr(|X_n-X|\ge \epsilon)<\epsilon_2$. Now comes my problem: Since this holds for all $n\ge N$, why is it then not equal to the alternative characterisation of a.s. convergence?
UPDATE: Is it correct to say that the difference is that if $\epsilon$ and $\epsilon_2$ is given. Than for all $n \ge N$ you can in the first case use the same subset of the sample-space. But in the case of only convergence in probability, you may have to change the subset of the sample space for each $n \ge N$?