It is the lemma 5.1 in this paper: https://dornsife.usc.edu/assets/sites/1193/docs/lin.pdf, and the paper contains the proof.
My attempt is:
By Portmanteau theorem, $X_n \to_d X$ is the equivalent as $\phi_{X_n}(t) \to_d \phi_X (t)$ for any $t \in \mathbb R$, so $X_n \to_d X$ implies that $\phi_{X_n}(t) \to \phi_X (t)$, and $X_n + Y_n \to_d X$ implies that $\phi_{X_n + Y_n}(t) = \phi_{X_n} (t) \phi_{Y_n} (t) \to \phi_X(t)$ for any $t \in \mathbb R$-- the equality holds by the independence of $X_n$ and $Y_n$. Therefore, it must be that $\phi_{Y_n} (t) \to 1 = E[e^{i t\cdot 0}]$ for any $t \in \mathbb R$, which implies that $Y_n \to_d 0$. Since the convergence in distribution to a constant value is equivalent as the convergence in probability to a constant value, it follows that $Y_n \to_p 0$.
I cannot see what in my proof might be wrong, but my proof concerns me, because it seems to be too simple compared to the one in the paper.