 The proof of the first statement, by Julius Worpitzky in 1865, is apparently the oldest published proof that a continued fraction with complex elements actually converges.Because the proof of Worpitzky's theorem employs Euler's continued fraction formula to construct an infinite series that is equivalent to the continued fraction "x", and the series so constructed is absolutely convergent, the Weierstrass M-test can be applied to a modified version of "x".If

and a positive real number "M" exists such that |"c""i"| ≤ "M" ("i" = 2, 3, 4, ...), then the sequence of convergents {"f""i"("z")} converges uniformly when

and "f"("z") is analytic on that open disk.