let X be RealUnitarySpace; :: thesis: for seq, seq9 being sequence of X st seq is convergent & ex k being Nat st

for n being Nat st k <= n holds

seq9 . n = seq . n holds

seq9 is convergent

let seq, seq9 be sequence of X; :: thesis: ( seq is convergent & ex k being Nat st

for n being Nat st k <= n holds

seq9 . n = seq . n implies seq9 is convergent )

assume that

A1: seq is convergent and

A2: ex k being Nat st

for n being Nat st k <= n holds

seq9 . n = seq . n ; :: thesis: seq9 is convergent

consider k being Nat such that

A3: for n being Nat st n >= k holds

seq9 . n = seq . n by A2;

consider g being Point of X such that

A4: for r being Real st r > 0 holds

ex m being Nat st

for n being Nat st n >= m holds

dist ((seq . n),g) < r by A1;

take h = g; :: according to BHSP_2:def 1 :: thesis: for r being Real st r > 0 holds

ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let r be Real; :: thesis: ( r > 0 implies ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

assume r > 0 ; :: thesis: ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

then consider m1 being Nat such that

A5: for n being Nat st n >= m1 holds

dist ((seq . n),h) < r by A4;

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r by A6; :: thesis: verum

for n being Nat st k <= n holds

seq9 . n = seq . n holds

seq9 is convergent

let seq, seq9 be sequence of X; :: thesis: ( seq is convergent & ex k being Nat st

for n being Nat st k <= n holds

seq9 . n = seq . n implies seq9 is convergent )

assume that

A1: seq is convergent and

A2: ex k being Nat st

for n being Nat st k <= n holds

seq9 . n = seq . n ; :: thesis: seq9 is convergent

consider k being Nat such that

A3: for n being Nat st n >= k holds

seq9 . n = seq . n by A2;

consider g being Point of X such that

A4: for r being Real st r > 0 holds

ex m being Nat st

for n being Nat st n >= m holds

dist ((seq . n),g) < r by A1;

take h = g; :: according to BHSP_2:def 1 :: thesis: for r being Real st r > 0 holds

ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let r be Real; :: thesis: ( r > 0 implies ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

assume r > 0 ; :: thesis: ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

then consider m1 being Nat such that

A5: for n being Nat st n >= m1 holds

dist ((seq . n),h) < r by A4;

A6: now :: thesis: ( m1 <= k implies ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

assume A7:
m1 <= k
; :: thesis: ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

take m = k; :: thesis: for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let n be Nat; :: thesis: ( n >= m implies dist ((seq9 . n),h) < r )

assume A8: n >= m ; :: thesis: dist ((seq9 . n),h) < r

then n >= m1 by A7, XXREAL_0:2;

then dist ((seq . n),g) < r by A5;

hence dist ((seq9 . n),h) < r by A3, A8; :: thesis: verum

end;for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

take m = k; :: thesis: for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let n be Nat; :: thesis: ( n >= m implies dist ((seq9 . n),h) < r )

assume A8: n >= m ; :: thesis: dist ((seq9 . n),h) < r

then n >= m1 by A7, XXREAL_0:2;

then dist ((seq . n),g) < r by A5;

hence dist ((seq9 . n),h) < r by A3, A8; :: thesis: verum

now :: thesis: ( k <= m1 implies ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

hence
ex m being Nat st for n being Nat st n >= m holds

dist ((seq9 . n),h) < r )

assume A9:
k <= m1
; :: thesis: ex m being Nat st

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

take m = m1; :: thesis: for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let n be Nat; :: thesis: ( n >= m implies dist ((seq9 . n),h) < r )

assume A10: n >= m ; :: thesis: dist ((seq9 . n),h) < r

then seq9 . n = seq . n by A3, A9, XXREAL_0:2;

hence dist ((seq9 . n),h) < r by A5, A10; :: thesis: verum

end;for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

take m = m1; :: thesis: for n being Nat st n >= m holds

dist ((seq9 . n),h) < r

let n be Nat; :: thesis: ( n >= m implies dist ((seq9 . n),h) < r )

assume A10: n >= m ; :: thesis: dist ((seq9 . n),h) < r

then seq9 . n = seq . n by A3, A9, XXREAL_0:2;

hence dist ((seq9 . n),h) < r by A5, A10; :: thesis: verum

for n being Nat st n >= m holds

dist ((seq9 . n),h) < r by A6; :: thesis: verum