let X be RealUnitarySpace; :: thesis: for seq1, seq2 being sequence of X st seq1 is convergent & seq2 is convergent holds
seq1 + seq2 is convergent
let seq1, seq2 be sequence of X; :: thesis: ( seq1 is convergent & seq2 is convergent implies seq1 + seq2 is convergent )
assume that
A1: seq1 is convergent and
A2: seq2 is convergent ; :: thesis: seq1 + seq2 is convergent
consider g1 being Point of X such that
A3: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
dist ((seq1 . n),g1) < r by A1;
consider g2 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 ((seq2 . n),g2) < r by A2;
take g = g1 + g2; :: 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 (((seq1 + seq2) . n),g) < r
let r be Real; :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
dist (((seq1 + seq2) . n),g) < r )
assume A5: r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
dist (((seq1 + seq2) . n),g) < r
then consider m1 being Nat such that
A6: for n being Nat st n >= m1 holds
dist ((seq1 . n),g1) < r / 2 by A3, XREAL_1:215;
consider m2 being Nat such that
A7: for n being Nat st n >= m2 holds
dist ((seq2 . n),g2) < r / 2 by A4, A5, XREAL_1:215;
reconsider k = m1 + m2 as Nat ;
take k ; :: thesis: for n being Nat st n >= k holds
dist (((seq1 + seq2) . n),g) < r
let n be Nat; :: thesis: ( n >= k implies dist (((seq1 + seq2) . n),g) < r )
assume A8: n >= k ; :: thesis: dist (((seq1 + seq2) . n),g) < r
k >= m2 by NAT_1:12;
then n >= m2 by A8, XXREAL_0:2;
then A9: dist ((seq2 . n),g2) < r / 2 by A7;
dist (((seq1 + seq2) . n),g) = dist (((seq1 . n) + (seq2 . n)),(g1 + g2)) by NORMSP_1:def 2;
then A10: dist (((seq1 + seq2) . n),g) <= (dist ((seq1 . n),g1)) + (dist ((seq2 . n),g2)) by BHSP_1:40;
m1 + m2 >= m1 by NAT_1:12;
then n >= m1 by A8, XXREAL_0:2;
then dist ((seq1 . n),g1) < r / 2 by A6;
then (dist ((seq1 . n),g1)) + (dist ((seq2 . n),g2)) < (r / 2) + (r / 2) by A9, XREAL_1:8;
hence dist (((seq1 + seq2) . n),g) < r by A10, XXREAL_0:2; :: thesis: verum