"The specific aim of the was to find an elementary proof of a special case of the density (DHJ)... The theorem was known to be true, but for mathematicians, proofs are more than guarantees of truth: they are valued for their explanatory power, and a new proof of a theorem can provide crucial insights. There were two reasons to want a new proof... First... [it] had just one—a long and complicated proof that relied on heavy mathematical machinery. An elementary proof—one that starts from first principles instead of relying on advanced techniques—would require many new ideas. Second, DHJ implies another famous theorem, called , novel proofs of which led to several breakthroughs over the past decade, so there was reason to expect that the same would happen..."