Complete convergence implies almost-sure convergence, with no independence needed

Let $\{X_n\}$ be a sequence of random variables and X a random variable such that $\sum_{n=1}^\infty P[|X_n-X|>\varepsilon]<\infty$ for every $\varepsilon>0$. Consider the statements: (I) $P[\lim_{n\to\infty}X_n=X]=1$. (II) $P[\lim_{n\to\infty}X_n^2=X^2]=1$. (III) $P[\lim_{n\to\infty}X_n=X]$ only if $\{X_n\}$ is a sequence of independent random variables.