= Solution
{c}
<Independent random variables> satisfy, for every finite selection of distinct indices $n_1,\ldots,n_m$ and <Borel sets> $C_1,\ldots,C_m$,
$$
\mathbb P\left(\bigcap_{j=1}^m\{X_{n_j}\in C_j\}\right)=\prod_{j=1}^m\mathbb P(X_{n_j}\in C_j).
$$
The <Second Borel-Cantelli lemma> says that if events $E_n$ are independent and $\sum_n\mathbb P(E_n)=\infty$, then
$$
\boxed{\mathbb P(\limsup_nE_n)=1.}
$$
Here $\limsup_nE_n=\bigcap_N\bigcup_{n\geq N}E_n$ is the event that infinitely many of the $E_n$ occur.
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