let f1, f2 be PartFunc of REAL,REAL; for x0 being Real st f1 is_right_divergent_to-infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_above ) holds
f1 + f2 is_right_divergent_to-infty_in x0
let x0 be Real; ( f1 is_right_divergent_to-infty_in x0 & ( for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 + f2) ) ) & ex r being Real st
( 0 < r & f2 | ].x0,(x0 + r).[ is bounded_above ) implies f1 + f2 is_right_divergent_to-infty_in x0 )
assume that
A1:
f1 is_right_divergent_to-infty_in x0
and
A2:
for r being Real st x0 < r holds
ex g being Real st
( g < r & x0 < g & g in dom (f1 + f2) )
; ( for r being Real holds
( not 0 < r or not f2 | ].x0,(x0 + r).[ is bounded_above ) or f1 + f2 is_right_divergent_to-infty_in x0 )
given r being Real such that A3:
0 < r
and
A4:
f2 | ].x0,(x0 + r).[ is bounded_above
; f1 + f2 is_right_divergent_to-infty_in x0
now for seq being Real_Sequence st seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) holds
(f1 + f2) /* seq is divergent_to-infty let seq be
Real_Sequence;
( seq is convergent & lim seq = x0 & rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0) implies (f1 + f2) /* seq is divergent_to-infty )assume that A5:
seq is
convergent
and A6:
lim seq = x0
and A7:
rng seq c= (dom (f1 + f2)) /\ (right_open_halfline x0)
;
(f1 + f2) /* seq is divergent_to-infty
x0 < x0 + r
by A3, Lm2;
then consider k being
Nat such that A8:
for
n being
Nat st
k <= n holds
seq . n < x0 + r
by A5, A6, LIMFUNC2:2;
A9:
(dom (f1 + f2)) /\ (right_open_halfline x0) c= dom (f1 + f2)
by XBOOLE_1:17;
A10:
rng (seq ^\ k) c= rng seq
by VALUED_0:21;
then A11:
rng (seq ^\ k) c= (dom (f1 + f2)) /\ (right_open_halfline x0)
by A7;
A12:
rng (seq ^\ k) c= dom (f1 + f2)
by A10, A7, A9;
dom (f1 + f2) = (dom f1) /\ (dom f2)
by VALUED_1:def 1;
then A13:
(
dom (f1 + f2) c= dom f1 &
dom (f1 + f2) c= dom f2 )
by XBOOLE_1:17;
then A14:
(
rng (seq ^\ k) c= dom f1 &
rng (seq ^\ k) c= dom f2 )
by A9, A11;
then
rng (seq ^\ k) c= (dom f1) /\ (dom f2)
by XBOOLE_1:19;
then A15:
(f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) =
(f1 + f2) /* (seq ^\ k)
by RFUNCT_2:8
.=
((f1 + f2) /* seq) ^\ k
by A7, A9, VALUED_0:27, XBOOLE_1:1
;
A16:
(dom (f1 + f2)) /\ (right_open_halfline x0) c= right_open_halfline x0
by XBOOLE_1:17;
then
rng (seq ^\ k) c= right_open_halfline x0
by A10, A7;
then A17:
rng (seq ^\ k) c= (dom f1) /\ (right_open_halfline x0)
by A14, XBOOLE_1:19;
now ex r2 being set st
for n being Nat holds (f2 /* (seq ^\ k)) . n < r2consider r1 being
Real such that A18:
for
g being
object st
g in ].x0,(x0 + r).[ /\ (dom f2) holds
f2 . g <= r1
by A4, RFUNCT_1:70;
take r2 =
r1 + 1;
for n being Nat holds (f2 /* (seq ^\ k)) . n < r2let n be
Nat;
(f2 /* (seq ^\ k)) . n < r2A19:
n in NAT
by ORDINAL1:def 12;
seq . (n + k) < x0 + r
by A8, NAT_1:12;
then A20:
(seq ^\ k) . n < x0 + r
by NAT_1:def 3;
A21:
(seq ^\ k) . n in rng (seq ^\ k)
by VALUED_0:28;
then
(seq ^\ k) . n in right_open_halfline x0
by A10, A7, A16;
then
(seq ^\ k) . n in { g1 where g1 is Real : x0 < g1 }
by XXREAL_1:230;
then
ex
g being
Real st
(
g = (seq ^\ k) . n &
x0 < g )
;
then
(seq ^\ k) . n in { g2 where g2 is Real : ( x0 < g2 & g2 < x0 + r ) }
by A20;
then
(seq ^\ k) . n in ].x0,(x0 + r).[
by RCOMP_1:def 2;
then
(seq ^\ k) . n in ].x0,(x0 + r).[ /\ (dom f2)
by A14, A21, XBOOLE_0:def 4;
then
f2 . ((seq ^\ k) . n) < r1 + 1
by A18, XREAL_1:39;
hence
(f2 /* (seq ^\ k)) . n < r2
by A12, A13, FUNCT_2:108, XBOOLE_1:1, A19;
verum end; then A22:
f2 /* (seq ^\ k) is
bounded_above
by SEQ_2:def 3;
lim (seq ^\ k) = x0
by A5, A6, SEQ_4:20;
then
(f1 /* (seq ^\ k)) + (f2 /* (seq ^\ k)) is
divergent_to-infty
by A22, A1, A5, A17, LIMFUNC2:def 6, LIMFUNC1:12;
hence
(f1 + f2) /* seq is
divergent_to-infty
by A15, LIMFUNC1:7;
verum end;
hence
f1 + f2 is_right_divergent_to-infty_in x0
by A2, LIMFUNC2:def 6; verum