Quadratic-mean convergence — what survives under addition, multiplication and $e^{i(\cdot)}$

Suppose $\{X_n\}$ and $\{Y_n\}$ are sequences of random variables and X and Y are random variables such that as $n\to\infty$, $X_n$ converges in quadratic mean to X and $Y_n$ converges in quadratic mean to Y. Consider the following statements: (I) The sequence $(X_n+Y_n)$ converges in quadratic mean to $(X+Y)$ as $n\to\infty$. (II) $\lim_{n\to\infty}P[|X_nY_n-XY|>\varepsilon]=0,\ \forall\varepsilon>0$. (III) $\lim_{n\to\infty}E[e^{iX_n}]=E[e^{iX}]$.