let X be RealUnitarySpace; :: thesis: for seq being sequence of X holds
( seq is convergent iff ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r )
let seq be sequence of X; :: thesis: ( seq is convergent iff ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r )
thus ( seq is convergent implies ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r ) :: thesis: ( ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r implies seq is convergent )

proof

assume seq is convergent ; :: thesis: ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r
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 g ; :: thesis: for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r
let r be Real; :: thesis: ( r > 0 implies ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r )
assume r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r
then consider m1 being Nat such that
A2: 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
||.((seq . n) - g).|| < r
let n be Nat; :: thesis: ( n >= k implies ||.((seq . n) - g).|| < r )
assume n >= k ; :: thesis: ||.((seq . n) - g).|| < r
then dist ((seq . n),g) < r by A2;
hence ||.((seq . n) - g).|| < r by BHSP_1:def 5; :: thesis: verum

end;

( ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r implies seq is convergent )

proof

given g 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
||.((seq . n) - g).|| < r ; :: thesis: seq is convergent
take 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 ((seq . 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 ((seq . n),g) < r )
assume r > 0 ; :: thesis: ex m being Nat st
for n being Nat st n >= m holds
dist ((seq . n),g) < r
then consider m1 being Nat such that
A4: for n being Nat st n >= m1 holds
||.((seq . n) - g).|| < r by A3;
take k = m1; :: thesis: for n being Nat st n >= k holds
dist ((seq . n),g) < r
let n be Nat; :: thesis: ( n >= k implies dist ((seq . n),g) < r )
assume n >= k ; :: thesis: dist ((seq . n),g) < r
then ||.((seq . n) - g).|| < r by A4;
hence dist ((seq . n),g) < r by BHSP_1:def 5; :: thesis: verum

end;

hence ( ex g being Point of X st
for r being Real st r > 0 holds
ex m being Nat st
for n being Nat st n >= m holds
||.((seq . n) - g).|| < r implies seq is convergent ) ; :: thesis: verum