consider N being Nat such that
A1: for n being Nat st n >= N holds
f . n >= 0 by Def2;
c + f is eventually-nonnegative

proof

take N ; :: according to ASYMPT_0:def 2 :: thesis: for n being Nat st n >= N holds
(c + f) . n >= 0
let n be Nat; :: thesis: ( n >= N implies (c + f) . n >= 0 )
A2: n in NAT by ORDINAL1:def 12;
assume n >= N ; :: thesis: (c + f) . n >= 0
then f . n >= 0 by A1;
then c + (f . n) >= 0 + 0 ;
hence (c + f) . n >= 0 by VALUED_1:2, A2; :: thesis: verum

end;

hence c + f is eventually-nonnegative Real_Sequence ; :: thesis: verum