
theorem Th86:
for seq be ExtREAL_sequence, r be R_eal st
 (for n be Nat holds seq.n <= r) holds lim_sup seq <= r
proof
   let seq be ExtREAL_sequence, r be R_eal;
   assume
A1: for n be Nat holds seq.n <= r;
   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 seq.n <= f.n
   proof
    let n be Nat;
    f.n = r by A4;
    hence seq.n <= f.n by A1;
   end; then
   lim_sup seq <= lim_sup f by MESFUN10:3;
   hence lim_sup seq <= r by A5,RINFSUP2:41;
end;
