>>11035283We construct the real numbers as the completion (under Cauchy sequences) of Q with the metric d(x,y) = |x-y|.
Define a Cauchy sequence {p_n} by the constant sequence {1, 1, ...} and another Cauchy sequence {q_n} by the sequence {0.9, 0.99, ...} (i.e., the partial sums of 0.999...).
Clearly |p_n - q_n| -> 0 for sufficiently large n, and so these Cauchy sequences are equivalent.
Because we have constructed R from Q as the set of equivalence classes of equivalent sequences, {p_n} and {q_n} both define the same real number, and so are equal.
