let X be set ; :: thesis: for r being Real st 0 < r & r < 1 holds
X -powers r is summable
let r be Real; :: thesis: ( 0 < r & r < 1 implies X -powers r is summable )
assume that
A1: 0 < r and
A2: r < 1 ; :: thesis: X -powers r is summable

A3: now :: thesis: for n being Nat holds 0 <= (X -powers r) . n

let n be Nat; :: thesis: 0 <= (X -powers r) . n
( ( n in X & (X -powers r) . n = r |^ n ) or ( not n in X & (X -powers r) . n = 0 ) ) by Def5;
hence 0 <= (X -powers r) . n by A1, PREPOWER:6; :: thesis: verum

end;

A4: now :: thesis: ex m being Nat st
for n being Nat st m <= n holds
(X -powers r) . n <= (r GeoSeq) . n

reconsider m = 1 as Nat ;
take m = m; :: thesis: for n being Nat st m <= n holds
(X -powers r) . n <= (r GeoSeq) . n
let n be Nat; :: thesis: ( m <= n implies (X -powers r) . n <= (r GeoSeq) . n )
assume m <= n ; :: thesis: (X -powers r) . n <= (r GeoSeq) . n
A5: (r GeoSeq) . n = r |^ n by PREPOWER:def 1;
( ( n in X & (X -powers r) . n = r |^ n ) or ( not n in X & (X -powers r) . n = 0 ) ) by Def5;
hence (X -powers r) . n <= (r GeoSeq) . n by A1, A5, PREPOWER:6; :: thesis: verum

end;

|.r.| = r by A1, ABSVALUE:def 1;
then r GeoSeq is summable by A2, SERIES_1:24;
hence X -powers r is summable by A4, A3, SERIES_1:19; :: thesis: verum