My professor just went over weak convergence in lecture and mentioned that there is a continuity requirement for the points of F which we require to converge. He offered this example as a way to see why this is required but I am having trouble working it through.
Consider a sequence of random variables with the pmf's $P(X_n = 1 + \frac{1}{n}) = 1, n \geq 1$, and $P(X = 1) = 1$, and let $F_n$ and $F$ be the associated cdf's. Show that $F_n => F$.