# Probability: Terminology Question for Convergence in Distribution

I'm currently probability from two different sources: the classic text by Billingsley and the course notes of an instructor at my university. I've run into a terminology conflict that I was hoping someone might clear up.

Let $F_1, F_2, ...$ be a sequence of distribution functions. According to Billingsley, $F_n$ is said to converge weakly to $F$ if (i) $F_n (x) \to F(x)$ for $x$ such that $F$ is continuous at $x$ and (ii) $F$ is also a distribution function.

Conversely, my notes say $F_n \to F$ weakly requires (i) but not (ii). If both (i) and (ii) are satisfied then we say $F_n \to F$ completely.

So, which is the more standard use? If I subscribe to Billingsley's definition, does complete convergence then have some different meaning [e.g. by strengthening (i)]?

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