MATH.21:5.2 - Construct an infinite word and return a finite observation

Let p_n be a binary word of length n, with p_n a prefix of p_(n+1). Positions start at zero. Define b(k) to be entry k of p_(k+1). Compatibility makes that entry agree with every later prefix, so b is an infinite binary word whose first n entries are p_n.

For a common space of approximations, pad each p_n with zeros to obtain an infinite word b_n. Use convergence by eventual agreement on each finite prefix. For a request about the first m entries, every b_n with n≥m agrees there with b. This proves convergence and gives the return stage: p_m supplies those entries. Any calculation depending only on them, such as their count of ones, can use that finite result.

The global property “has only finitely many ones” does not pass to this limit. Take p_n to consist of n ones. Each zero-padded b_n has finitely many ones, but b has a one at every position. A question about a fixed finite prefix is settled; a question about the entire tail needs another argument.

The construction uses a consistency relation between finite stages and a rule for obtaining every entry. Its computational realization additionally needs a way to obtain the required p_m. Merely knowing that such prefixes exist does not supply their generator.