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

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

let n be Nat; :: thesis: ( 0 <= (X -powers r) . n & (X -powers r) . n <= (r GeoSeq) . n )
A4: ( ( 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: (X -powers r) . n <= (r GeoSeq) . n
(r GeoSeq) . n = r |^ n by PREPOWER:def 1;
hence (X -powers r) . n <= (r GeoSeq) . n by A1, A4, PREPOWER:6; :: thesis: verum

end;

|.r.| = r by A1, ABSVALUE:def 1;
then r GeoSeq is summable by A2, SERIES_1:24;
hence Sum (X -powers r) <= Sum (r GeoSeq) by A3, SERIES_1:20; :: thesis: verum