On the convergence of sequences in the space of n-iterated function systems with applications

The paper’s Theorem 3.13 states that a minimally ordered, eventually decreasing, Cauchy sequence of hyperbolic n-IFSs on a complete (X,d) converges in (S^n_X, D), building on the assignment metric D and the d̄∞ metric on maps, and using the fact that an eventually decreasing Cauchy sequence of contractions converges to a contraction (Theorem 3.5) . The candidate solution proves the same result by (i) verifying the metric properties of d̄∞ and D consistent with the paper’s definitions and proofs , (ii) using the minimally ordered condition to telescope along consecutive pairs and show coordinatewise Cauchy-ness, and (iii) using eventual decrease of contractivity factors to conclude each limit is a contraction, hence convergence in D. This mirrors the paper’s structure (Cauchy+eventual decrease ⇒ limit contraction; then D-convergence), with more explicit tail/telescoping estimates. One minor note: the paper’s Proposition 3.12 proof that (C(X), d̄∞) is complete is correct in conclusion but a bit terse about the uniform convergence step; the candidate gives a clean completeness argument for X^X that also suffices. Overall, both are correct and essentially the same proof strategy, with the candidate filling in more details.