See also Convergence of measures

https://www.wikiwand.com/en/Proofs_of_convergence_of_random_variables#/propA1i

https://math.stackexchange.com/questions/149775/convergence-of-random-variables-in-probability-but-not-almost-surely <- we want to show that the probability that I go to $1$ infinitely often (a sequence $X_i$ is said to visit $1$ infinitely often if for any $n_0$ I can always find an $n>n_0$ where $X_n=1$ in the sequence) is $1$. Uses Borel-Cantelli lemmas