let x0 be Real; :: thesis: for f being PartFunc of REAL,REAL holds
( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) )
let f be PartFunc of REAL,REAL; :: thesis: ( f is_divergent_to-infty_in x0 iff ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) )
thus ( f is_divergent_to-infty_in x0 implies ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 ) ) :: thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 implies f is_divergent_to-infty_in x0 )
proof
assume A1: f is_divergent_to-infty_in x0 ; :: thesis: ( f is_left_divergent_to-infty_in x0 & f is_right_divergent_to-infty_in x0 )
A2: now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) holds
f /* s is divergent_to-infty
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (left_open_halfline x0) implies f /* s is divergent_to-infty )
assume that
A3: s is convergent and
A4: lim s = x0 and
A5: rng s c= (dom f) /\ (left_open_halfline x0) ; :: thesis: f /* s is divergent_to-infty
rng s c= (dom f) \ {x0} by A5, Th1;
hence f /* s is divergent_to-infty by A1, A3, A4; :: thesis: verum
end;
A6: now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) holds
f /* s is divergent_to-infty
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) /\ (right_open_halfline x0) implies f /* s is divergent_to-infty )
assume that
A7: s is convergent and
A8: lim s = x0 and
A9: rng s c= (dom f) /\ (right_open_halfline x0) ; :: thesis: f /* s is divergent_to-infty
rng s c= (dom f) \ {x0} by A9, Th1;
hence f /* s is divergent_to-infty by A1, A7, A8; :: thesis: verum
end;
A10: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A1;
then for r being Real st r < x0 holds
ex g being Real st
( r < g & g < x0 & g in dom f ) by Th8;
hence f is_left_divergent_to-infty_in x0 by A2, LIMFUNC2:def 3; :: thesis: f 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 f ) by A10, Th8;
hence f is_right_divergent_to-infty_in x0 by A6, LIMFUNC2:def 6; :: thesis: verum
end;
assume that
A11: f is_left_divergent_to-infty_in x0 and
A12: f is_right_divergent_to-infty_in x0 ; :: thesis: f is_divergent_to-infty_in x0
A13: now :: thesis: for s being Real_Sequence st s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} holds
f /* s is divergent_to-infty
let s be Real_Sequence; :: thesis: ( s is convergent & lim s = x0 & rng s c= (dom f) \ {x0} implies f /* s is divergent_to-infty )
assume that
A14: s is convergent and
A15: lim s = x0 and
A16: rng s c= (dom f) \ {x0} ; :: thesis: f /* s is divergent_to-infty
now :: thesis: f /* s is divergent_to-infty
per cases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
s . n < x0 or for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n >= x0 ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
s . n < x0 ; :: thesis: f /* s is divergent_to-infty
then consider k being Element of NAT such that
A17: for n being Element of NAT st k <= n holds
s . n < x0 ;
A18: rng s c= dom f by A16, XBOOLE_1:1;
A19: rng (s ^\ k) c= (dom f) /\ (left_open_halfline x0)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (left_open_halfline x0) )
assume x in rng (s ^\ k) ; :: thesis: x in (dom f) /\ (left_open_halfline x0)
then consider n being Element of NAT such that
A20: (s ^\ k) . n = x by FUNCT_2:113;
s . (n + k) < x0 by A17, NAT_1:12;
then s . (n + k) in { g1 where g1 is Real : g1 < x0 } ;
then s . (n + k) in left_open_halfline x0 by XXREAL_1:229;
then A21: x in left_open_halfline x0 by A20, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A20, NAT_1:def 3;
hence x in (dom f) /\ (left_open_halfline x0) by A18, A21, XBOOLE_0:def 4; :: thesis: verum
end;
A22: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A14, A15, SEQ_4:20;
then f /* (s ^\ k) is divergent_to-infty by A11, A14, A19, LIMFUNC2:def 3;
hence f /* s is divergent_to-infty by A22, LIMFUNC1:7; :: thesis: verum
end;
suppose A23: for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n >= x0 ) ; :: thesis: f /* s is divergent_to-infty
now :: thesis: f /* s is divergent_to-infty
per cases ( ex k being Element of NAT st
for n being Element of NAT st k <= n holds
x0 < s . n or for k being Element of NAT ex n being Element of NAT st
( k <= n & x0 >= s . n ) ) ;
suppose ex k being Element of NAT st
for n being Element of NAT st k <= n holds
x0 < s . n ; :: thesis: f /* s is divergent_to-infty
then consider k being Element of NAT such that
A24: for n being Element of NAT st k <= n holds
s . n > x0 ;
A25: rng s c= dom f by A16, XBOOLE_1:1;
A26: rng (s ^\ k) c= (dom f) /\ (right_open_halfline x0)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s ^\ k) or x in (dom f) /\ (right_open_halfline x0) )
assume x in rng (s ^\ k) ; :: thesis: x in (dom f) /\ (right_open_halfline x0)
then consider n being Element of NAT such that
A27: (s ^\ k) . n = x by FUNCT_2:113;
x0 < s . (n + k) by A24, NAT_1:12;
then s . (n + k) in { g1 where g1 is Real : x0 < g1 } ;
then s . (n + k) in right_open_halfline x0 by XXREAL_1:230;
then A28: x in right_open_halfline x0 by A27, NAT_1:def 3;
s . (n + k) in rng s by VALUED_0:28;
then x in rng s by A27, NAT_1:def 3;
hence x in (dom f) /\ (right_open_halfline x0) by A25, A28, XBOOLE_0:def 4; :: thesis: verum
end;
A29: f /* (s ^\ k) = (f /* s) ^\ k by A16, VALUED_0:27, XBOOLE_1:1;
lim (s ^\ k) = x0 by A14, A15, SEQ_4:20;
then f /* (s ^\ k) is divergent_to-infty by A12, A14, A26, LIMFUNC2:def 6;
hence f /* s is divergent_to-infty by A29, LIMFUNC1:7; :: thesis: verum
end;
suppose A30: for k being Element of NAT ex n being Element of NAT st
( k <= n & x0 >= s . n ) ; :: thesis: f /* s is divergent_to-infty
defpred S1[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds
( n < m & s . m < x0 & ( for k being Element of NAT st n < k & s . k < x0 holds
m <= k ) );
defpred S2[ Nat, set , set ] means S1[$2,$3];
defpred S3[ Nat] means s . $1 < x0;
A31: now :: thesis: for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n < x0 )
let k be Element of NAT ; :: thesis: ex n being Element of NAT st
( k <= n & s . n < x0 )
consider n being Element of NAT such that
A32: k <= n and
A33: s . n <= x0 by A30;
take n = n; :: thesis: ( k <= n & s . n < x0 )
thus k <= n by A32; :: thesis: s . n < x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A16, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n < x0 by A33, XXREAL_0:1; :: thesis: verum
end;
then ex m1 being Element of NAT st
( 0 <= m1 & s . m1 < x0 ) ;
then A34: ex m being Nat st S3[m] ;
consider M being Nat such that
A35: ( S3[M] & ( for n being Nat st S3[n] holds
M <= n ) ) from NAT_1:sch 5(A34);
reconsider M9 = M as Element of NAT by ORDINAL1:def 12;
A36: now :: thesis: for n being Element of NAT ex m being Element of NAT st
( n < m & s . m < x0 )
let n be Element of NAT ; :: thesis: ex m being Element of NAT st
( n < m & s . m < x0 )
consider m being Element of NAT such that
A37: n + 1 <= m and
A38: s . m < x0 by A31;
take m = m; :: thesis: ( n < m & s . m < x0 )
thus ( n < m & s . m < x0 ) by A37, A38, NAT_1:13; :: thesis: verum
end;
A39: for n being Nat
for x being Element of NAT ex y being Element of NAT st S2[n,x,y]
proof
let n be Nat; :: thesis: for x being Element of NAT ex y being Element of NAT st S2[n,x,y]
let x be Element of NAT ; :: thesis: ex y being Element of NAT st S2[n,x,y]
defpred S4[ Nat] means ( x < $1 & s . $1 < x0 );
ex m being Element of NAT st S4[m] by A36;
then A40: ex m being Nat st S4[m] ;
consider l being Nat such that
A41: ( S4[l] & ( for k being Nat st S4[k] holds
l <= k ) ) from NAT_1:sch 5(A40);
 take l ; :: thesis: ( l is Element of NAT & S2[n,x,l] )
l in NAT by ORDINAL1:def 12;
hence ( l is Element of NAT & S2[n,x,l] ) by A41; :: thesis: verum
end;
consider F being sequence of NAT such that
A42: ( F . 0 = M9 & ( for n being Nat holds S2[n,F . n,F . (n + 1)] ) ) from RECDEF_1:sch 2(A39);
A43: rng F c= NAT by RELAT_1:def 19;
then A44: rng F c= REAL by NUMBERS:19;
A45: dom F = NAT by FUNCT_2:def 1;
then reconsider F = F as Real_Sequence by A44, RELSET_1:4;
A46: now :: thesis: for n being Element of NAT holds F . n is Element of NAT
let n be Element of NAT ; :: thesis: F . n is Element of NAT
F . n in rng F by A45, FUNCT_1:def 3;
hence F . n is Element of NAT by A43; :: thesis: verum
end;
now :: thesis: for n being Nat holds F . n < F . (n + 1)
let n be Nat; :: thesis: F . n < F . (n + 1)
A47: n in NAT by ORDINAL1:def 12;
A48: F . (n + 1) is Element of NAT by A46;
F . n is Element of NAT by A46, A47;
hence F . n < F . (n + 1) by A42, A48; :: thesis: verum
end;
then reconsider F = F as increasing sequence of NAT by SEQM_3:def 6;
A49: s * F is subsequence of s by VALUED_0:def 17;
then rng (s * F) c= rng s by VALUED_0:21;
then A50: rng (s * F) c= (dom f) \ {x0} by A16;
defpred S4[ Nat] means ( s . $1 < x0 & ( for m being Element of NAT holds F . m <> $1 ) );
A51: for n being Element of NAT st s . n < x0 holds
ex m being Element of NAT st F . m = n
proof
assume ex n being Element of NAT st S4[n] ; :: thesis: contradiction
then A52: ex n being Nat st S4[n] ;
consider M1 being Nat such that
A53: ( S4[M1] & ( for n being Nat st S4[n] holds
M1 <= n ) ) from NAT_1:sch 5(A52);
 defpred S5[ Nat] means ( $1 < M1 & s . $1 < x0 & ex m being Element of NAT st F . m = $1 );
A54: ex n being Nat st S5[n]
proof
take M ; :: thesis: S5[M]
A55: M <> M1 by A42, A53;
M <= M1 by A35, A53;
hence M < M1 by A55, XXREAL_0:1; :: thesis: ( s . M < x0 & ex m being Element of NAT st F . m = M )
thus s . M < x0 by A35; :: thesis: ex m being Element of NAT st F . m = M
take 0 ; :: thesis: F . 0 = M
thus F . 0 = M by A42; :: thesis: verum
end;
A56: for n being Nat st S5[n] holds
n <= M1 ;
consider MX being Nat such that
A57: ( S5[MX] & ( for n being Nat st S5[n] holds
n <= MX ) ) from NAT_1:sch 6(A56, A54);
 A58: for k being Element of NAT st MX < k & k < M1 holds
s . k >= x0
proof
given k being Element of NAT such that A59: MX < k and
A60: k < M1 and
A61: s . k < x0 ; :: thesis: contradiction
now :: thesis: contradiction
per cases ( ex m being Element of NAT st F . m = k or for m being Element of NAT holds F . m <> k ) ;
end;
end;
hence contradiction ; :: thesis: verum
end;
consider m being Element of NAT such that
A62: F . m = MX by A57;
M1 in NAT by ORDINAL1:def 12;
then A63: F . (m + 1) <= M1 by A42, A53, A57, A62;
A64: s . (F . (m + 1)) < x0 by A42, A62;
A65: MX < F . (m + 1) by A42, A62;
now :: thesis: not F . (m + 1) <> M1
assume F . (m + 1) <> M1 ; :: thesis: contradiction
then F . (m + 1) < M1 by A63, XXREAL_0:1;
hence contradiction by A58, A65, A64; :: thesis: verum
end;
hence contradiction by A53; :: thesis: verum
end;
defpred S5[ Nat] means s . $1 > x0;
A66: now :: thesis: for k being Element of NAT ex n being Element of NAT st
( k <= n & s . n > x0 )
let k be Element of NAT ; :: thesis: ex n being Element of NAT st
( k <= n & s . n > x0 )
consider n being Element of NAT such that
A67: k <= n and
A68: s . n >= x0 by A23;
take n = n; :: thesis: ( k <= n & s . n > x0 )
thus k <= n by A67; :: thesis: s . n > x0
s . n in rng s by VALUED_0:28;
then not s . n in {x0} by A16, XBOOLE_0:def 5;
then s . n <> x0 by TARSKI:def 1;
hence s . n > x0 by A68, XXREAL_0:1; :: thesis: verum
end;
then ex mn being Element of NAT st
( 0 <= mn & s . mn > x0 ) ;
then A69: ex m being Nat st S5[m] ;
consider N being Nat such that
A70: ( S5[N] & ( for n being Nat st S5[n] holds
N <= n ) ) from NAT_1:sch 5(A69);
defpred S6[ Nat] means (s * F) . $1 < x0;
A71: for k being Nat st S6[k] holds
S6[k + 1]
proof
let k be Nat; :: thesis: ( S6[k] implies S6[k + 1] )
assume (s * F) . k < x0 ; :: thesis: S6[k + 1]
S1[F . k,F . (k + 1)] by A42;
then s . (F . (k + 1)) < x0 ;
hence S6[k + 1] by FUNCT_2:15; :: thesis: verum
end;
A72: S6[ 0 ] by A35, A42, FUNCT_2:15;
A73: for k being Nat holds S6[k] from NAT_1:sch 2(A72, A71);
A74: rng (s * F) c= (dom f) /\ (left_open_halfline x0)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s * F) or x in (dom f) /\ (left_open_halfline x0) )
assume A75: x in rng (s * F) ; :: thesis: x in (dom f) /\ (left_open_halfline x0)
then consider n being Element of NAT such that
A76: (s * F) . n = x by FUNCT_2:113;
(s * F) . n < x0 by A73;
then x in { g1 where g1 is Real : g1 < x0 } by A76;
then A77: x in left_open_halfline x0 by XXREAL_1:229;
x in dom f by A50, A75, XBOOLE_0:def 5;
hence x in (dom f) /\ (left_open_halfline x0) by A77, XBOOLE_0:def 4; :: thesis: verum
end;
defpred S7[ set , set ] means for n, m being Element of NAT st $1 = n & $2 = m holds
( n < m & s . m > x0 & ( for k being Element of NAT st n < k & s . k > x0 holds
m <= k ) );
defpred S8[ Nat, set , set ] means S7[$2,$3];
A78: s * F is convergent by A14, A49, SEQ_4:16;
lim (s * F) = x0 by A14, A15, A49, SEQ_4:17;
then A79: f /* (s * F) is divergent_to-infty by A11, A78, A74, LIMFUNC2:def 3;
reconsider N9 = N as Element of NAT by ORDINAL1:def 12;
A80: now :: thesis: for n being Element of NAT ex m being Element of NAT st
( n < m & s . m > x0 )
let n be Element of NAT ; :: thesis: ex m being Element of NAT st
( n < m & s . m > x0 )
consider m being Element of NAT such that
A81: n + 1 <= m and
A82: s . m > x0 by A66;
take m = m; :: thesis: ( n < m & s . m > x0 )
thus ( n < m & s . m > x0 ) by A81, A82, NAT_1:13; :: thesis: verum
end;
A83: for n being Nat
for x being Element of NAT ex y being Element of NAT st S8[n,x,y]
proof
let n be Nat; :: thesis: for x being Element of NAT ex y being Element of NAT st S8[n,x,y]
let x be Element of NAT ; :: thesis: ex y being Element of NAT st S8[n,x,y]
defpred S9[ Nat] means ( x < $1 & s . $1 > x0 );
ex m being Element of NAT st S9[m] by A80;
then A84: ex m being Nat st S9[m] ;
consider l being Nat such that
A85: ( S9[l] & ( for k being Nat st S9[k] holds
l <= k ) ) from NAT_1:sch 5(A84);
 take l ; :: thesis: ( l is Element of NAT & S8[n,x,l] )
l in NAT by ORDINAL1:def 12;
hence ( l is Element of NAT & S8[n,x,l] ) by A85; :: thesis: verum
end;
consider G being sequence of NAT such that
A86: ( G . 0 = N9 & ( for n being Nat holds S8[n,G . n,G . (n + 1)] ) ) from RECDEF_1:sch 2(A83);
A87: rng G c= NAT by RELAT_1:def 19;
then A88: rng G c= REAL by NUMBERS:19;
A89: dom G = NAT by FUNCT_2:def 1;
then reconsider G = G as Real_Sequence by A88, RELSET_1:4;
A90: now :: thesis: for n being Element of NAT holds G . n is Element of NAT
let n be Element of NAT ; :: thesis: G . n is Element of NAT
G . n in rng G by A89, FUNCT_1:def 3;
hence G . n is Element of NAT by A87; :: thesis: verum
end;
now :: thesis: for n being Nat holds G . n < G . (n + 1)
let n be Nat; :: thesis: G . n < G . (n + 1)
A91: n in NAT by ORDINAL1:def 12;
A92: G . (n + 1) is Element of NAT by A90;
G . n is Element of NAT by A90, A91;
hence G . n < G . (n + 1) by A86, A92; :: thesis: verum
end;
then reconsider G = G as increasing sequence of NAT by SEQM_3:def 6;
A93: s * G is subsequence of s by VALUED_0:def 17;
then rng (s * G) c= rng s by VALUED_0:21;
then A94: rng (s * G) c= (dom f) \ {x0} by A16;
defpred S9[ Nat] means ( s . $1 > x0 & ( for m being Element of NAT holds G . m <> $1 ) );
A95: for n being Element of NAT st s . n > x0 holds
ex m being Element of NAT st G . m = n
proof
assume ex n being Element of NAT st S9[n] ; :: thesis: contradiction
then A96: ex n being Nat st S9[n] ;
consider N1 being Nat such that
A97: ( S9[N1] & ( for n being Nat st S9[n] holds
N1 <= n ) ) from NAT_1:sch 5(A96);
 defpred S10[ Nat] means ( $1 < N1 & s . $1 > x0 & ex m being Element of NAT st G . m = $1 );
A98: ex n being Nat st S10[n]
proof
take N ; :: thesis: S10[N]
A99: N <> N1 by A86, A97;
N <= N1 by A70, A97;
hence N < N1 by A99, XXREAL_0:1; :: thesis: ( s . N > x0 & ex m being Element of NAT st G . m = N )
thus s . N > x0 by A70; :: thesis: ex m being Element of NAT st G . m = N
take 0 ; :: thesis: G . 0 = N
thus G . 0 = N by A86; :: thesis: verum
end;
A100: for n being Nat st S10[n] holds
n <= N1 ;
consider NX being Nat such that
A101: ( S10[NX] & ( for n being Nat st S10[n] holds
n <= NX ) ) from NAT_1:sch 6(A100, A98);
 A102: for k being Element of NAT st NX < k & k < N1 holds
s . k <= x0
proof
given k being Element of NAT such that A103: NX < k and
A104: k < N1 and
A105: s . k > x0 ; :: thesis: contradiction
now :: thesis: contradiction
per cases ( ex m being Element of NAT st G . m = k or for m being Element of NAT holds G . m <> k ) ;
end;
end;
hence contradiction ; :: thesis: verum
end;
consider m being Element of NAT such that
A106: G . m = NX by A101;
N1 in NAT by ORDINAL1:def 12;
then A107: G . (m + 1) <= N1 by A86, A97, A101, A106;
A108: s . (G . (m + 1)) > x0 by A86, A106;
A109: NX < G . (m + 1) by A86, A106;
now :: thesis: not G . (m + 1) <> N1
assume G . (m + 1) <> N1 ; :: thesis: contradiction
then G . (m + 1) < N1 by A107, XXREAL_0:1;
hence contradiction by A102, A109, A108; :: thesis: verum
end;
hence contradiction by A97; :: thesis: verum
end;
defpred S10[ Nat] means (s * G) . $1 > x0;
A110: for k being Nat st S10[k] holds
S10[k + 1]
proof
let k be Nat; :: thesis: ( S10[k] implies S10[k + 1] )
assume (s * G) . k > x0 ; :: thesis: S10[k + 1]
S7[G . k,G . (k + 1)] by A86;
then s . (G . (k + 1)) > x0 ;
hence S10[k + 1] by FUNCT_2:15; :: thesis: verum
end;
A111: S10[ 0 ] by A70, A86, FUNCT_2:15;
A112: for k being Nat holds S10[k] from NAT_1:sch 2(A111, A110);
A113: rng (s * G) c= (dom f) /\ (right_open_halfline x0)
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (s * G) or x in (dom f) /\ (right_open_halfline x0) )
assume A114: x in rng (s * G) ; :: thesis: x in (dom f) /\ (right_open_halfline x0)
then consider n being Element of NAT such that
A115: (s * G) . n = x by FUNCT_2:113;
(s * G) . n > x0 by A112;
then x in { g1 where g1 is Real : x0 < g1 } by A115;
then A116: x in right_open_halfline x0 by XXREAL_1:230;
x in dom f by A94, A114, XBOOLE_0:def 5;
hence x in (dom f) /\ (right_open_halfline x0) by A116, XBOOLE_0:def 4; :: thesis: verum
end;
A117: s * G is convergent by A14, A93, SEQ_4:16;
lim (s * G) = x0 by A14, A15, A93, SEQ_4:17;
then A118: f /* (s * G) is divergent_to-infty by A12, A117, A113, LIMFUNC2:def 6;
now :: thesis: for r being Real ex n being Nat st
for k being Nat st n <= k holds
(f /* s) . k < r
let r be Real; :: thesis: ex n being Nat st
for k being Nat st n <= k holds
(f /* s) . k < r
consider n1 being Nat such that
A119: for k being Nat st n1 <= k holds
(f /* (s * F)) . k < r by A79;
consider n2 being Nat such that
A120: for k being Nat st n2 <= k holds
(f /* (s * G)) . k < r by A118;
reconsider n = max ((F . n1),(G . n2)) as Nat ;
take n = n; :: thesis: for k being Nat st n <= k holds
(f /* s) . k < r
let k be Nat; :: thesis: ( n <= k implies (f /* s) . k < r )
A121: k in NAT by ORDINAL1:def 12;
assume A122: n <= k ; :: thesis: (f /* s) . k < r
s . k in rng s by VALUED_0:28;
then not s . k in {x0} by A16, XBOOLE_0:def 5;
then A123: s . k <> x0 by TARSKI:def 1;
now :: thesis: (f /* s) . k < r
per cases ( s . k < x0 or s . k > x0 ) by A123, XXREAL_0:1;
suppose s . k < x0 ; :: thesis: (f /* s) . k < r
then consider l being Element of NAT such that
A124: k = F . l by A51, A121;
F . n1 <= n by XXREAL_0:25;
then F . n1 <= k by A122, XXREAL_0:2;
then l >= n1 by A124, SEQM_3:1;
then (f /* (s * F)) . l < r by A119;
then f . ((s * F) . l) < r by A50, FUNCT_2:108, XBOOLE_1:1;
then f . (s . k) < r by A124, FUNCT_2:15;
hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1, A121; :: thesis: verum
end;
suppose s . k > x0 ; :: thesis: (f /* s) . k < r
then consider l being Element of NAT such that
A125: k = G . l by A95, A121;
G . n2 <= n by XXREAL_0:25;
then G . n2 <= k by A122, XXREAL_0:2;
then l >= n2 by A125, SEQM_3:1;
then (f /* (s * G)) . l < r by A120;
then f . ((s * G) . l) < r by A94, FUNCT_2:108, XBOOLE_1:1;
then f . (s . k) < r by A125, FUNCT_2:15;
hence (f /* s) . k < r by A16, FUNCT_2:108, XBOOLE_1:1, A121; :: thesis: verum
end;
end;
end;
hence (f /* s) . k < r ; :: thesis: verum
end;
hence f /* s is divergent_to-infty ; :: thesis: verum
end;
end;
end;
hence f /* s is divergent_to-infty ; :: thesis: verum
end;
end;
end;
hence f /* s is divergent_to-infty ; :: thesis: verum
end;
now :: thesis: for r1, r2 being Real st r1 < x0 & x0 < r2 holds
ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
let r1, r2 be Real; :: thesis: ( r1 < x0 & x0 < r2 implies ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) )
assume that
A126: r1 < x0 and
A127: x0 < r2 ; :: thesis: ex g1, g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
consider g1 being Real such that
A128: r1 < g1 and
A129: g1 < x0 and
A130: g1 in dom f by A11, A126, LIMFUNC2:def 3;
consider g2 being Real such that
A131: g2 < r2 and
A132: x0 < g2 and
A133: g2 in dom f by A12, A127, LIMFUNC2:def 6;
take g1 = g1; :: thesis: ex g2 being Real st
( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
take g2 = g2; :: thesis: ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f )
thus ( r1 < g1 & g1 < x0 & g1 in dom f & g2 < r2 & x0 < g2 & g2 in dom f ) by A128, A129, A130, A131, A132, A133; :: thesis: verum
end;
hence f is_divergent_to-infty_in x0 by A13; :: thesis: verum