
theorem Th10:
for f be PartFunc of REAL,REAL, a be Real st
 f is convergent_in-infty & f is non-increasing holds
  for x be Real st x in dom f holds f.x <= lim_in-infty f
proof
    let f be PartFunc of REAL,REAL, a be Real;
    assume that
A1:  f is convergent_in-infty and
A2:  f is non-increasing;

    let x be Real;
    assume
A3:  x in dom f;

    hereby assume
A4:  f.x > lim_in-infty f; then
A5:  f.x - lim_in-infty f > 0 by XREAL_1:50;
     set g1 = f.x - lim_in-infty f;
     consider r be Real such that
A6:   for r1 be Real st r1 < r & r1 in dom f holds
       |. f.r1 - lim_in-infty f .| < g1 by A5,A1,LIMFUNC1:78;

     consider R be Real such that
A7:   R < min(x,r) & R in dom f by A1,LIMFUNC1:45;
A8: x >= min(x,r) & r >= min(x,r) by XXREAL_0:17; then
     R < r by A7,XXREAL_0:2; then
A9: |. f.R - lim_in-infty f .| < g1 by A6,A7;

     x > R by A8,A7,XXREAL_0:2; then
A10:  f.x <= f.R by A2,A3,A7,RFUNCT_2:def 4; then
     lim_in-infty f < f.R by A4,XXREAL_0:2; then
     f.R - lim_in-infty f > 0 by XREAL_1:50; then
     f.R - lim_in-infty f < g1 by A9,ABSVALUE:def 1;
     hence contradiction by A10,XREAL_1:9;
    end;
end;
