let X be ComplexUnitarySpace; :: thesis: for x being Point of X
for seq being sequence of X st seq is constant & x in rng seq holds
lim seq = x
let x be Point of X; :: thesis: for seq being sequence of X st seq is constant & x in rng seq holds
lim seq = x
let seq be sequence of X; :: thesis: ( seq is constant & x in rng seq implies lim seq = x )
assume that
A1: seq is constant and
A2: x in rng seq ; :: thesis: lim seq = x
consider y being Point of X such that
A3: rng seq = {y} by A1, FUNCT_2:111;
consider w being Point of X such that
A4: for n being Nat holds seq . n = w by A1, VALUED_0:def 18;
A5: x = y by A2, A3, TARSKI:def 1;
A6: now :: thesis: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq . n),x) < r
let r be Real; :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
dist ((seq . n),x) < r )
assume A7: r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
dist ((seq . n),x) < r
reconsider m = 0 as Nat ;
take m = m; :: thesis: for n being Nat st n >= m holds
dist ((seq . n),x) < r
let n be Nat; :: thesis: ( n >= m implies dist ((seq . n),x) < r )
assume n >= m ; :: thesis: dist ((seq . n),x) < r
n in NAT by ORDINAL1:def 12;
then n in dom seq by NORMSP_1:12;
then seq . n in rng seq by FUNCT_1:def 3;
then w in rng seq by A4;
then w = x by A3, A5, TARSKI:def 1;
then seq . n = x by A4;
hence dist ((seq . n),x) < r by A7, CSSPACE:50; :: thesis: verum
end;
seq is convergent by A1, Th1;
hence lim seq = x by A6, Def2; :: thesis: verum