THE CLOSURE OF CONVERGENCE SETS FOR CONTINUED FRACTIONS ARE CONVERGENCE SETS

We prove that if OMEGA is a simple convergence set for continued fractions K(a(n)/b(n)), then the closure OMEGABAR of OMEGA is also such a convergence set. Actually, we prove more: every continued fraction K(a(n)/b(n)) has a ''neighbourhood'' {D(n)}infinity/n=1; D(n)={z is-an-element-of C; \z-a(n)\less-than-or-equal-to r(n)} x {z is-an-element-of C; \z-b(n)\less-than-or-equal-to s(n)} where r(n)>0 and s(n)>0, with the following property: Every continued fraction from {D(n)} converges if and only if K(a(n)/b(n)) converges.