let s, s1 be Real_Sequence; :: thesis: ( ( for n being Nat holds s1 . n = n -root ((abs s) . n) ) & ex m being Nat st
for n being Nat st m <= n holds
s1 . n >= 1 implies not s is summable )
assume A1: for n being Nat holds s1 . n = n -root ((abs s) . n) ; :: thesis: ( for m being Nat ex n being Nat st
( m <= n & not s1 . n >= 1 ) or not s is summable )
given m being Nat such that A2: for n being Nat st m <= n holds
s1 . n >= 1 ; :: thesis: not s is summable
now :: thesis: for n being Nat st n >= m + 1 holds
|.(s . n).| >= 1
let n be Nat; :: thesis: ( n >= m + 1 implies |.(s . n).| >= 1 )
assume A3: n >= m + 1 ; :: thesis: |.(s . n).| >= 1
m + 1 >= 1 by NAT_1:11;
then A4: n >= 1 by A3, XXREAL_0:2;
m + 1 >= m by NAT_1:11;
then A5: n >= m by A3, XXREAL_0:2;
s1 . n = n -root ((abs s) . n) by A1
.= n -root |.(s . n).| by SEQ_1:12 ;
then ( |.(s . n).| >= 0 & (n -root |.(s . n).|) |^ n >= 1 ) by A2, A5, COMPLEX1:46, PREPOWER:11;
hence |.(s . n).| >= 1 by A4, POWER:4; :: thesis: verum
end;
then ( not s is convergent or lim s <> 0 ) by Th38;
hence not s is summable by Th4; :: thesis: verum