  We give a modern account of Agafonov's original proof of his eponymous
theorem. The original proof was only reported in Russian in a journal not
widely available, and the work most commonly cited in western literature is
instead the English translation of a summary version containing no proofs, and
the main proof relied heavily on material well-known in Russian mathematical
circles of the day, which perhaps obscures the main thrust of argumentation for
modern readers.Our present account recasts Aganofov's arguments using more
basic building blocks than in the original proof, and contains some further
embellishments to Agafonov's original arguments, made in the interest of
clarity. We posit that the modern account provides new insight to the
underlying phenomena of the theorem.We also provides some historical context to
Agafonov's work, including a short description of some of the ideas that led to
Agafonov's own proof, especially emphasizing the important work of Postnikova.
