Cofiniteness is $Σ_3^0$-complete How can we prove that the language $COF=\{M\mid L(M)\text{ is cofinite}\}$ is m-complete for the class $ Σ_3^0$ ?
I have shown that $COF\inΣ_3^0$ but I can't prove that $COF$ is m-hard for $Σ_3^0$ which means that every other problem $L\in Σ_3^0$ satisfies $L\leq_m COF$.
Your problem is not so trivial, though classical. See the proof of Corollary 11.9 in lecture notes of Kevin T. Kelly.