let seq be ExtREAL_sequence; :: thesis: ( seq is convergent_to_+infty implies ( not seq is convergent_to_-infty & not seq is convergent_to_finite_number ) )

assume A1: seq is convergent_to_+infty ; :: thesis: ( not seq is convergent_to_-infty & not seq is convergent_to_finite_number )

then consider g being Real such that

A4: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - g).| < p ;

reconsider g1 = g as R_eal by XXREAL_0:def 1;

assume A1: seq is convergent_to_+infty ; :: thesis: ( not seq is convergent_to_-infty & not seq is convergent_to_finite_number )

hereby :: thesis: not seq is convergent_to_finite_number

assume
seq is convergent_to_finite_number
; :: thesis: contradictionassume
seq is convergent_to_-infty
; :: thesis: contradiction

then consider n1 being Nat such that

A2: for m being Nat st n1 <= m holds

seq . m <= - 1 ;

consider n2 being Nat such that

A3: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) <= - 1 by A2, XXREAL_0:25;

hence contradiction by A3, XXREAL_0:25; :: thesis: verum

end;then consider n1 being Nat such that

A2: for m being Nat st n1 <= m holds

seq . m <= - 1 ;

consider n2 being Nat such that

A3: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) <= - 1 by A2, XXREAL_0:25;

hence contradiction by A3, XXREAL_0:25; :: thesis: verum

then consider g being Real such that

A4: for p being Real st 0 < p holds

ex n being Nat st

for m being Nat st n <= m holds

|.((seq . m) - g).| < p ;

reconsider g1 = g as R_eal by XXREAL_0:def 1;

per cases
( g > 0 or g = 0 or g < 0 )
;

end;

suppose A5:
g > 0
; :: thesis: contradiction

then consider n1 being Nat such that

A6: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < g by A4;

A8: for m being Nat st n2 <= m holds

2 * g <= seq . m by A1, A5;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) < 2 * g by A7, XXREAL_0:25;

hence contradiction by A8, XXREAL_0:25; :: thesis: verum

end;A6: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < g by A4;

A7: now :: thesis: for m being Nat st n1 <= m holds

seq . m < 2 * g

consider n2 being Nat such that seq . m < 2 * g

let m be Nat; :: thesis: ( n1 <= m implies seq . m < 2 * g )

assume n1 <= m ; :: thesis: seq . m < 2 * g

then |.((seq . m) - g).| < g by A6;

then (seq . m) - g1 < g by EXTREAL1:21;

then seq . m < g + g by XXREAL_3:54;

hence seq . m < 2 * g ; :: thesis: verum

end;assume n1 <= m ; :: thesis: seq . m < 2 * g

then |.((seq . m) - g).| < g by A6;

then (seq . m) - g1 < g by EXTREAL1:21;

then seq . m < g + g by XXREAL_3:54;

hence seq . m < 2 * g ; :: thesis: verum

A8: for m being Nat st n2 <= m holds

2 * g <= seq . m by A1, A5;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) < 2 * g by A7, XXREAL_0:25;

hence contradiction by A8, XXREAL_0:25; :: thesis: verum

suppose A9:
g = 0
; :: thesis: contradiction

consider n1 being Nat such that

A10: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < 1 by A4;

consider n2 being Nat such that

A11: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

reconsider jj = 1 as R_eal by XXREAL_0:def 1;

set m = max (n1,n2);

|.((seq . (max (n1,n2))) - g1).| < jj by A10, XXREAL_0:25;

then (seq . (max (n1,n2))) - g1 < jj by EXTREAL1:21;

then seq . (max (n1,n2)) < 1 + g by XXREAL_3:54;

then seq . (max (n1,n2)) < 1 by A9;

hence contradiction by A11, XXREAL_0:25; :: thesis: verum

end;A10: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < 1 by A4;

consider n2 being Nat such that

A11: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

reconsider jj = 1 as R_eal by XXREAL_0:def 1;

set m = max (n1,n2);

|.((seq . (max (n1,n2))) - g1).| < jj by A10, XXREAL_0:25;

then (seq . (max (n1,n2))) - g1 < jj by EXTREAL1:21;

then seq . (max (n1,n2)) < 1 + g by XXREAL_3:54;

then seq . (max (n1,n2)) < 1 by A9;

hence contradiction by A11, XXREAL_0:25; :: thesis: verum

suppose A12:
g < 0
; :: thesis: contradiction

consider n1 being Nat such that

A13: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < - g1 by A4, A12;

A15: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) < 0 by A14, XXREAL_0:25;

hence contradiction by A15, XXREAL_0:25; :: thesis: verum

end;A13: for m being Nat st n1 <= m holds

|.((seq . m) - g).| < - g1 by A4, A12;

A14: now :: thesis: for m being Element of NAT st n1 <= m holds

seq . m < 0

consider n2 being Nat such that seq . m < 0

let m be Element of NAT ; :: thesis: ( n1 <= m implies seq . m < 0 )

assume n1 <= m ; :: thesis: seq . m < 0

then |.((seq . m) - g1).| < - g1 by A13;

then (seq . m) - g1 < - g1 by EXTREAL1:21;

then seq . m < g - g1 by XXREAL_3:54;

hence seq . m < 0 by XXREAL_3:7; :: thesis: verum

end;assume n1 <= m ; :: thesis: seq . m < 0

then |.((seq . m) - g1).| < - g1 by A13;

then (seq . m) - g1 < - g1 by EXTREAL1:21;

then seq . m < g - g1 by XXREAL_3:54;

hence seq . m < 0 by XXREAL_3:7; :: thesis: verum

A15: for m being Nat st n2 <= m holds

1 <= seq . m by A1;

reconsider n1 = n1, n2 = n2 as Element of NAT by ORDINAL1:def 12;

set m = max (n1,n2);

seq . (max (n1,n2)) < 0 by A14, XXREAL_0:25;

hence contradiction by A15, XXREAL_0:25; :: thesis: verum