Cauchy sequence
For some authors, notably Royden, the Riesz-Fischer Theorem is the result that formula_1 is complete: that every Cauchy sequence of functions in formula_1 converges to a function in formula_52 under the metric induced by the "p"-norm. The proof below is based on the convergence theorems for the Lebesgue integral; the result can also be obtained for formula_53 by showing that every Cauchy sequence has a rapidly converging Cauchy sub-sequence, that every Cauchy sequence with a convergent sub-sequence converges, and that every rapidly Cauchy sequence in formula_1 converges in formula_55