
theorem Th87:
for seq be ExtREAL_sequence, r be R_eal st
 (for n be Nat holds r <= seq.n) holds r <= lim_inf seq
proof
   let seq be ExtREAL_sequence, r be R_eal;
   assume
A1: for n be Nat holds r <= seq.n;
   deffunc F(Element of NAT) = r;
   consider f be Function of NAT,ExtREAL such that
A2: for n be Element of NAT holds f.n = F(n) from FUNCT_2:sch 4;
A4:for n be Nat holds f.n = r
   proof
    let n be Nat;
    n is Element of NAT by ORDINAL1:def 12;
    hence f.n = r by A2;
   end; then
A5:f is convergent & lim f = r by MESFUNC5:60;
   for n be Nat holds f.n <= seq.n
   proof
    let n be Nat;
    f.n = r by A4;
    hence f.n <= seq.n by A1;
   end; then
   lim_inf f <= lim_inf seq by MESFUN10:3;
   hence r <= lim_inf seq by A5,RINFSUP2:41;
end;
