co 3. Strong Convergence and Cycle-Free Power Series Proof implies 37 If r is cycle-free then, by definition, lim (r, B), B)"=O. Conversely, if n-'Y implies lim (r, B)"=O. Hence r is cycle-free. 6 n--+-I 0 The following notational convention for power series in (A {l'n) {l'n will be applied in the sequel. If r E (A {l'n) {l'n then we write sometimes r= I [(r, v)] v instead of r= I (r, v) D.

Then we have I-I(IlM,j)= n kEC(j) (1-1(11, k»)c;: kEC(j) This implies that, for all i E I-I(IlM,}) and} E I, Hence, (02) is satisfied. 6, the choice being now I(IlM,})=C(J) uI(Il,}), for all}EI. 10. The mapping limc: Dc ~ A~ x I defined by (limcll)i,j=limlli,j, 11 E Dc, i,) E I, is a limit function on Dc, 4. 7. 49 0 Again, the next theorem summarizes our main result. 11. Assume that lim: D ~ A is a limit function. Then also the mapping lime: De ~ A~ x1 defined by (lime l1)i. ' with A1xI R . 12.

Furthermore, show that lim: D ---+