let X be ComplexUnitarySpace; :: thesis: for seq being sequence of X
for r being Real st r > 0 & ex m being Nat st
for n being Nat st n >= m holds
||.(seq . n).|| >= r & seq is convergent holds
lim seq <> 0. X
let seq be sequence of X; :: thesis: for r being Real st r > 0 & ex m being Nat st
for n being Nat st n >= m holds
||.(seq . n).|| >= r & seq is convergent holds
lim seq <> 0. X
let r be Real; :: thesis: ( r > 0 & ex m being Nat st
for n being Nat st n >= m holds
||.(seq . n).|| >= r & seq is convergent implies lim seq <> 0. X )
assume A1: r > 0 ; :: thesis: ( for m being Nat ex n being Nat st
( n >= m & not ||.(seq . n).|| >= r ) or not seq is convergent or lim seq <> 0. X )
given m being Nat such that A2: for n being Nat st n >= m holds
||.(seq . n).|| >= r ; :: thesis: ( not seq is convergent or lim seq <> 0. X )
now :: thesis: ( not seq is convergent or lim seq <> 0. X )
per cases ( not seq is convergent or seq is convergent ) ;
suppose A3: seq is convergent ; :: thesis: ( not seq is convergent or lim seq <> 0. X )
now :: thesis: not lim seq = H1(X)
assume lim seq = H1(X) ; :: thesis: contradiction
then consider k being Nat such that
A4: for n being Nat st n >= k holds
||.((seq . n) - H1(X)).|| < r by A1, A3, CLVECT_2:19;
now :: thesis: for n being Nat holds not n >= m + k
let n be Nat; :: thesis: not n >= m + k
assume A5: n >= m + k ; :: thesis: contradiction
m + k >= k by NAT_1:11;
then n >= k by A5, XXREAL_0:2;
then ||.((seq . n) - H1(X)).|| < r by A4;
then A6: ||.(seq . n).|| < r by RLVECT_1:13;
m + k >= m by NAT_1:11;
then n >= m by A5, XXREAL_0:2;
hence contradiction by A2, A6; :: thesis: verum
end;
hence contradiction ; :: thesis: verum
end;
hence ( not seq is convergent or lim seq <> 0. X ) ; :: thesis: verum
end;
end;
end;
hence ( not seq is convergent or lim seq <> 0. X ) ; :: thesis: verum