
theorem Th5:
for f be PartFunc of REAL,REAL st f is convergent_in-infty holds
  (ex r be Real st f|left_open_halfline r is bounded_below) &
  (ex r be Real st f|left_open_halfline r is bounded_above)
proof
    let f be PartFunc of REAL,REAL;
    assume f is convergent_in-infty; then
    consider g be Real such that
A1:  for g1 be Real st 0 < g1 ex r be Real st for r1 be Real st
       r1<r & r1 in dom f holds |.f.r1-g.| < g1 by LIMFUNC1:45;
    consider r be Real such that
A2:  for r1 be Real st r1<r & r1 in dom f holds |.f.r1-g.| < 1 by A1;

    for r1 be object st r1 in dom(f|left_open_halfline r)
     holds -1+g < (f|left_open_halfline r).r1
    proof
     let r1 be object;
     assume A3: r1 in dom(f|left_open_halfline r); then
     reconsider r1 as Real;
     r1 in dom f /\ left_open_halfline r by A3,RELAT_1:61; then
A4:  r1 in dom f & r1 in left_open_halfline r by XBOOLE_0:def 4; then
     |. f.r1 - g .| < 1 by A2,XXREAL_1:233; then
A5:  -1 <= f.r1 - g by ABSVALUE:5;
     now assume -1 = f.r1 - g; then
      |. f.r1 - g .| = -(-1) by ABSVALUE:def 1;
      hence contradiction by A2,A4,XXREAL_1:233;
     end; then
     -1 < f.r1-g by A5,XXREAL_0:1; then
     -1+g < f.r1 by XREAL_1:20;
     hence thesis by A4,FUNCT_1:49;
    end; then
    f|left_open_halfline r is bounded_below by SEQ_2:def 2;
    hence ex r be Real st f|left_open_halfline r is bounded_below;

    consider r be Real such that
A6:  for r1 be Real st r1<r & r1 in dom f holds |.f.r1-g.| < 1 by A1;

    for r1 be object st r1 in dom(f|left_open_halfline r)
     holds (f|left_open_halfline r).r1 < g+1
    proof
     let r1 be object;
     assume A7: r1 in dom(f|left_open_halfline r); then
     reconsider r1 as Real;
     r1 in dom f /\ left_open_halfline r by A7,RELAT_1:61; then
A8:  r1 in dom f & r1 in left_open_halfline r by XBOOLE_0:def 4; then
     |. f.r1 - g .| < 1 by A6,XXREAL_1:233; then
     f.r1-g < 1 by ABSVALUE:def 1; then
     f.r1 < g+1 by XREAL_1:19;
     hence thesis by A8,FUNCT_1:49;
    end; then
    f|left_open_halfline r is bounded_above by SEQ_2:def 1;
    hence ex r be Real st f|left_open_halfline r is bounded_above;
end;
