Although of course both examples are perfectly fine. I just think the first is a better illustration of the desired result.

I think it's much more intuitive, because we can more easily confirm that X_n = 1 must always happen again, no matter where we are. In the 1/n example, we have to accept that there will be infinitely many additional 1's, even when the probability reaches arbitrarily small values

So this fulfills convergence in probability “perfectly” in my opinion, since X_n = 1 clearly becomes much less likely, but it’s equally clear that it will ALWAYS happen again. Compared to the usual example of P(X_n = 1) = 1/n, which can also easily be verified to converge in p but not a.s.,...

The idea behind convergence in probability is that it becomes increasingly unlikely that X_n > epsilon, while almost sure convergence states that X_n > epsilon is impossible past a certain point.