= Solution
{c}
By <convergence in probability>, choose recursively $n_k>n_{k-1}$ so that
$$
\mathbb P(|X_{n_k}-X|>2^{-k})<2^{-k}.
$$
The corresponding <events> have a summable sequence of probabilities. The <First Borel-Cantelli lemma>, which requires no independence, therefore makes the probability of infinitely many of them zero. On the complementary event, $|X_{n_k}-X|\leq2^{-k}$ for all sufficiently large $k$, proving
$$
\boxed{X_{n_k}\longrightarrow X\quad\text{almost surely}.}
$$
The recursively chosen indices are strictly increasing and hence tend to infinity.
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