let r be Real; :: thesis: for seq being Real_Sequence st seq is convergent holds
r (#) seq is convergent
let seq be Real_Sequence; :: thesis: ( seq is convergent implies r (#) seq is convergent )
assume seq is convergent ; :: thesis: r (#) seq is convergent
then consider g1 being Real such that
A1: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.((seq . m) - g1).| < p ;
take g = r * g1; :: according to SEQ_2:def 6 :: thesis: for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((r (#) seq) . m) - g).| < p

A2: now :: thesis: ( r = 0 implies for p being Real st 0 < p holds
ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p )

assume A3: r = 0 ; :: thesis: for p being Real st 0 < p holds
ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
let p be Real; :: thesis: ( 0 < p implies ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p )
assume A4: 0 < p ; :: thesis: ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
reconsider k = 0 as Nat ;
take k = k; :: thesis: for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
let m be Nat; :: thesis: ( k <= m implies |.(((r (#) seq) . m) - g).| < p )
assume k <= m ; :: thesis: |.(((r (#) seq) . m) - g).| < p
|.(((r (#) seq) . m) - g).| = |.((0 * (seq . m)) - 0).| by A3, SEQ_1:9
.= 0 by ABSVALUE:2 ;
hence |.(((r (#) seq) . m) - g).| < p by A4; :: thesis: verum

end;

now :: thesis: ( r <> 0 implies for p being Real st 0 < p holds
ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p )

assume A5: r <> 0 ; :: thesis: for p being Real st 0 < p holds
ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
then A6: 0 < |.r.| by COMPLEX1:47;
let p be Real; :: thesis: ( 0 < p implies ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p )
assume A7: 0 < p ; :: thesis: ex k being Nat st
for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
A8: 0 <> |.r.| by A5, COMPLEX1:47;
consider n1 being Nat such that
A9: for m being Nat st n1 <= m holds
|.((seq . m) - g1).| < p / |.r.| by A1, A6, A7;
take k = n1; :: thesis: for m being Nat st k <= m holds
|.(((r (#) seq) . m) - g).| < p
let m be Nat; :: thesis: ( k <= m implies |.(((r (#) seq) . m) - g).| < p )
assume k <= m ; :: thesis: |.(((r (#) seq) . m) - g).| < p
then A10: |.((seq . m) - g1).| < p / |.r.| by A9;
A11: |.(((r (#) seq) . m) - g).| = |.((r * (seq . m)) - (r * g1)).| by SEQ_1:9
.= |.(r * ((seq . m) - g1)).|
.= |.r.| * |.((seq . m) - g1).| by COMPLEX1:65 ;
|.r.| * (p / |.r.|) = |.r.| * ((|.r.| ") * p) by XCMPLX_0:def 9
.= (|.r.| * (|.r.| ")) * p
.= 1 * p by A8, XCMPLX_0:def 7
.= p ;
hence |.(((r (#) seq) . m) - g).| < p by A6, A10, A11, XREAL_1:68; :: thesis: verum

end;

hence for p being Real st 0 < p holds
ex n being Nat st
for m being Nat st n <= m holds
|.(((r (#) seq) . m) - g).| < p by A2; :: thesis: verum