Scaling a fixed random variable — convergence is to $X$, not to $0$

Suppose $\{X_n,n\ge1\}$ is a sequence of random variables defined as $X_n=\left(1+\frac1n\right)X$, where X is a discrete random variable with support $\{0,1\}$ and $P[X=0]=\frac23$. Following are three statements: (I) $X_n\to X$ in probability. (II) $X_n\to X$ in law. (III) $X_n\to X$ in quadratic mean.