Let ${xi_n}_{n geq 1} : (Omega, mathcal{F}, P) rightarrow (mathbb{R}^1, Bor)$ be a sequence of r.v.
Please, help me find connections between the following types of convergence as $n rightarrow infty$:

$xi_n xrightarrow{sLip} xi$, i.e. $Sigma_{n geq 1} Ef(xi_n) – f(xi) < infty$ $forall f in Lip$ and bounded

$xi_n xrightarrow{sdistr} xi$, i.e. $Sigma_{n geq 1} EF_{xi_n}(x) – F_{xi}(x) < infty$ $forall$ point $x$ of continuity of (distribution) $F_{xi}$.
I know that in special cases (for example when $f$ is $arctan$) the second convergence follows from the first convergence.
But I don’t know how to prove it in general (and find a counterexample that first convergence does not follows from the second, if I understand this correctly).