Thanks for the answer! I am trying to understand your proof of 1). I dont understand the sentence "But since $X_n$ is equal to one or zero..." and the steps after this. Can you explain it to me in more detail please?

X_n takes only two values, 0 and 1. So $P(X_n>ε)=P(X_n=1)$.

Thats the thing I don't understand. I am sorry for this probably stupid question. But why $X_n$ does only take two values?

Probably then you have just copied the exercise from somewhere then, without having read it! It is in the text that you posted, first line top right... $P(X_n=0)$ and $P(X_n=1)$. No other values.

But what about the $p_n$?I thought this notation does mean: $P(X_n=0)=P(X_n^{-1}(0))=p_n$ I thought thats just a property on a certain set and we don't know anything about the other values. I am reading it like: "The measure of the fiber under all $X_n$ of zero is exactly $p_n$"

It does not fit that you are doing probability theory with convergence, Borel-Cantelli etc, and you do not know basic notation. What you are asking is elementary notation. I am sorry.

Okay, i got it know. Last question:Why are you only considering $X=0$ and not also $X=1$? If all $X_n=1$ or $1$, then X could be 1 too.

The exercise says "show that X goes to 0 if.... Why should I consider that X goes to 1?? Are you making fun of me?

Hello

No, I am not doing fun of you

and i actually don't want to disturb you

i am just new to probability theory

So, for task 1). If we suppose the left hand side: X_n -> 0. Can i conclude, that:

Okay, i understood the implication A=>B

Now i have to understand B=>A (A is the statement on the left hand side, and B is the statement on the right hand side of task 1) )

Good luck dude