let X be RealUnitarySpace; :: thesis: for a being Real

for seq being sequence of X st seq is convergent holds

a * seq is convergent

let a be Real; :: thesis: for seq being sequence of X st seq is convergent holds

a * seq is convergent

let seq be sequence of X; :: thesis: ( seq is convergent implies a * seq is convergent )

assume seq is convergent ; :: thesis: a * seq is convergent

then consider g being Point of X such that

A1: 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 ;

take h = a * 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 (((a * seq) . n),h) < r

ex m being Nat st

for n being Nat st n >= m holds

dist (((a * seq) . n),h) < r by A2; :: thesis: verum

for seq being sequence of X st seq is convergent holds

a * seq is convergent

let a be Real; :: thesis: for seq being sequence of X st seq is convergent holds

a * seq is convergent

let seq be sequence of X; :: thesis: ( seq is convergent implies a * seq is convergent )

assume seq is convergent ; :: thesis: a * seq is convergent

then consider g being Point of X such that

A1: 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 ;

take h = a * 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 (((a * seq) . n),h) < r

A2: now :: thesis: ( a <> 0 implies for r being Real st r > 0 holds

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

A3:
0 / |.a.| = 0
;

assume A4: a <> 0 ; :: thesis: for r being Real st r > 0 holds

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then A5: |.a.| > 0 by COMPLEX1:47;

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then consider m1 being Nat such that

A6: for n being Nat st n >= m1 holds

dist ((seq . n),g) < r / |.a.| by A1, A5, A3, XREAL_1:74;

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

dist (((a * seq) . n),h) < r

let n be Nat; :: thesis: ( n >= k implies dist (((a * seq) . n),h) < r )

assume n >= k ; :: thesis: dist (((a * seq) . n),h) < r

then A7: dist ((seq . n),g) < r / |.a.| by A6;

A8: |.a.| <> 0 by A4, COMPLEX1:47;

A9: |.a.| * (r / |.a.|) = |.a.| * ((|.a.| ") * r) by XCMPLX_0:def 9

.= (|.a.| * (|.a.| ")) * r

.= 1 * r by A8, XCMPLX_0:def 7

.= r ;

dist ((a * (seq . n)),(a * g)) = ||.((a * (seq . n)) - (a * g)).|| by BHSP_1:def 5

.= ||.(a * ((seq . n) - g)).|| by RLVECT_1:34

.= |.a.| * ||.((seq . n) - g).|| by BHSP_1:27

.= |.a.| * (dist ((seq . n),g)) by BHSP_1:def 5 ;

then dist ((a * (seq . n)),h) < r by A5, A7, A9, XREAL_1:68;

hence dist (((a * seq) . n),h) < r by NORMSP_1:def 5; :: thesis: verum

end;assume A4: a <> 0 ; :: thesis: for r being Real st r > 0 holds

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then A5: |.a.| > 0 by COMPLEX1:47;

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then consider m1 being Nat such that

A6: for n being Nat st n >= m1 holds

dist ((seq . n),g) < r / |.a.| by A1, A5, A3, XREAL_1:74;

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

dist (((a * seq) . n),h) < r

let n be Nat; :: thesis: ( n >= k implies dist (((a * seq) . n),h) < r )

assume n >= k ; :: thesis: dist (((a * seq) . n),h) < r

then A7: dist ((seq . n),g) < r / |.a.| by A6;

A8: |.a.| <> 0 by A4, COMPLEX1:47;

A9: |.a.| * (r / |.a.|) = |.a.| * ((|.a.| ") * r) by XCMPLX_0:def 9

.= (|.a.| * (|.a.| ")) * r

.= 1 * r by A8, XCMPLX_0:def 7

.= r ;

dist ((a * (seq . n)),(a * g)) = ||.((a * (seq . n)) - (a * g)).|| by BHSP_1:def 5

.= ||.(a * ((seq . n) - g)).|| by RLVECT_1:34

.= |.a.| * ||.((seq . n) - g).|| by BHSP_1:27

.= |.a.| * (dist ((seq . n),g)) by BHSP_1:def 5 ;

then dist ((a * (seq . n)),h) < r by A5, A7, A9, XREAL_1:68;

hence dist (((a * seq) . n),h) < r by NORMSP_1:def 5; :: thesis: verum

now :: thesis: ( a = 0 implies for r being Real st r > 0 holds

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

hence
for r being Real st r > 0 holds ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

assume A10:
a = 0
; :: thesis: for r being Real st r > 0 holds

ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then consider m1 being Nat such that

A11: for n being Nat st n >= m1 holds

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

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

dist (((a * seq) . n),h) < r

let n be Nat; :: thesis: ( n >= k implies dist (((a * seq) . n),h) < r )

assume n >= k ; :: thesis: dist (((a * seq) . n),h) < r

then A12: dist ((seq . n),g) < r by A11;

dist ((a * (seq . n)),(a * g)) = dist ((0 * (seq . n)),H_{1}(X))
by A10, RLVECT_1:10

.= dist (H_{1}(X),H_{1}(X))
by RLVECT_1:10

.= 0 by BHSP_1:34 ;

then dist ((a * (seq . n)),h) < r by A12, BHSP_1:37;

hence dist (((a * seq) . n),h) < r by NORMSP_1:def 5; :: thesis: verum

end;ex k being Nat st

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r )

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

for n being Nat st n >= k holds

dist (((a * seq) . n),h) < r

then consider m1 being Nat such that

A11: for n being Nat st n >= m1 holds

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

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

dist (((a * seq) . n),h) < r

let n be Nat; :: thesis: ( n >= k implies dist (((a * seq) . n),h) < r )

assume n >= k ; :: thesis: dist (((a * seq) . n),h) < r

then A12: dist ((seq . n),g) < r by A11;

dist ((a * (seq . n)),(a * g)) = dist ((0 * (seq . n)),H

.= dist (H

.= 0 by BHSP_1:34 ;

then dist ((a * (seq . n)),h) < r by A12, BHSP_1:37;

hence dist (((a * seq) . n),h) < r by NORMSP_1:def 5; :: thesis: verum

ex m being Nat st

for n being Nat st n >= m holds

dist (((a * seq) . n),h) < r by A2; :: thesis: verum