let r be Real; :: thesis: for X being RealUnitarySpace
for seq being sequence of X 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 X be RealUnitarySpace; :: thesis: for seq being sequence of X 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: ( 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 )

per cases ( not seq is convergent or seq is convergent ) ;

suppose not seq is convergent ; :: thesis: ( not seq is convergent or lim seq <> 0. X )

hence ( not seq is convergent or lim seq <> 0. X ) ; :: thesis: verum

end;

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, BHSP_2:19;
set n = m + k;
m + k >= k by NAT_1:11;
then m + k >= k ;
then ||.((seq . (m + k)) - H₁(X)).|| < r by A4;
then A5: ||.(seq . (m + k)).|| < r ;
m + k >= m by NAT_1:11;
then m + k >= m ;
hence contradiction by A2, A5; :: thesis: verum

end;

hence ( not seq is convergent or lim seq <> 0. X ) ; :: thesis: verum

end;

end;