
theorem Th6:
for f be PartFunc of REAL,REAL st f is convergent_in+infty holds
  (ex r be Real st f|right_open_halfline r is bounded_below) &
  (ex r be Real st f|right_open_halfline r is bounded_above)
proof
    let f be PartFunc of REAL,REAL;
    assume
A1:  f is convergent_in+infty;
    consider g be Real such that
A2:  for g1 be Real st 0 < g1 ex r be Real st for r1 be Real st
       r<r1 & r1 in dom f holds |.f.r1-g.| < g1 by A1,LIMFUNC1:44;

    consider r be Real such that
A3:  for r1 be Real st r<r1 & r1 in dom f holds |.f.r1-g.| < 1 by A2;

    for r1 be object st r1 in dom(f|right_open_halfline r)
     holds -1+g < (f|right_open_halfline r).r1
    proof
     let r1 be object;
     assume A4: r1 in dom(f|right_open_halfline r); then
     reconsider r1 as Real;
     r1 in dom f /\ right_open_halfline r by A4,RELAT_1:61; then
A5:  r1 in dom f & r1 in right_open_halfline r by XBOOLE_0:def 4; then
     |. f.r1 - g .| < 1 by A3,XXREAL_1:235; then
A6:  -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 A3,A5,XXREAL_1:235;
     end; then
     -1 < f.r1-g by A6,XXREAL_0:1; then
     -1+g < f.r1 by XREAL_1:20;
     hence thesis by A5,FUNCT_1:49;
    end; then
    f|right_open_halfline r is bounded_below by SEQ_2:def 2;
    hence ex r be Real st f|right_open_halfline r is bounded_below;

    consider r be Real such that
A7:  for r1 be Real st r<r1 & r1 in dom f holds |.f.r1-g.| < 1 by A2;

    for r1 be object st r1 in dom(f|right_open_halfline r)
     holds (f|right_open_halfline r).r1 < g+1
    proof
     let r1 be object;
     assume A8: r1 in dom(f|right_open_halfline r); then
     reconsider r1 as Real;
     r1 in dom f /\ right_open_halfline r by A8,RELAT_1:61; then
A9:  r1 in dom f & r1 in right_open_halfline r by XBOOLE_0:def 4; then
     |. f.r1 - g .| < 1 by A7,XXREAL_1:235; then
     f.r1-g < 1 by ABSVALUE:def 1; then
     f.r1 < g+1 by XREAL_1:19;
     hence thesis by A9,FUNCT_1:49;
    end; then
    f|right_open_halfline r is bounded_above by SEQ_2:def 1;
    hence ex r be Real st f|right_open_halfline r is bounded_above;
end;
