Please, help me find connections between two types of convergence:

Let $\{X_n\}_{n\ge1}: (\Omega,F,P) \rightarrow (\mathbb{R},Bor)$ be a sequence of r.v., there are two convergences:

1. $X_n \rightarrow X \hspace{0.2cm}(sLip)$, i.e. $\sum_{n\ge1} E|f(X_n) - f(X)| < \infty \hspace{0.2cm}\forall f \in Lip$ and bounded

2. $X_n \rightarrow X \hspace{0.2cm}(c.c.)$, i.e. $\sum_{n\ge1} P(|X_n - X|\le\epsilon) = \infty \hspace{0.2cm}\forall \epsilon > 0$

I know several things about other type of "complete convergence" ($X_n \rightarrow X \hspace{0.2cm}(c.c.)$, i.e. $\sum_{n\ge1} P(|X_n - X|\le\epsilon) < \infty \hspace{0.2cm}\forall \epsilon > 0$) and it's connection with "strong $L^p$" convergence ($X_n \rightarrow X \hspace{0.2cm}(s.-L^p)$, i.e. $\sum_{n\ge1} E(|X_n - X|^p) < \infty$).

Also, I now about the second Borell-Cantelli lemma, but it uses the independence of random variables (which we don't have).

And it is easy to prove that $E|f(\xi_n)-f(\xi)| \le L E|\xi_n - \xi| \le L ||\xi_n - \xi||_{\infty}$ for L-Lipschitz and bounded functions.

But I don't know, how can I apply all these facts to the given situation (or maybe there is another way to solve this problem). If you have any ideas (or some articles to recommend), I will be very pleasant.