, Suppose that the sequence f d(?n,s) does not tend to 0. There exists a sequence (u m ) m? IN of IN * such that f d(?u m ,s) > b ?m ? IN

, Consequently, inf n?L f d(?n,s) = 0 and therefore f belongs to Ker(? s ), which proves that Ker(? s ) = Ker(? r )

, f q , ) of ? r , where f 1 , ..., f q ? A and > 0, Consider now a neighborhood V(? r , f 1

.. .. V(?-r-,-f-1, f q , ) = {? ? M ult(A, . ) | ? r (f j ) ? ?f j ) ?

?. , ,. .. , and ?. In-*-}, By definition of that topology, there exists a subset G of IN such that |? r (f j )? f j d(?n,r) | ? ? ?n ? G, ?j = 1, ..., q. But now, in each disk d(? n , r), we can take a class d(b n , r ? ) where none of the f j admits a zero, and hence we have |f j (b n )| = f j d(?n,r) ?j = 1, ..., q and hence |? bn (f j ) ? ? r (f j )| ? ? ?j = 1, ..., q. Consequently, ? bn belongs to V(? r , f 1

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