Let $φ_1,φ_2,\ldots$ and $ψ$ be sentences. Let $T$ be set of sentences $$\{φ_2\rightarrow φ_1,φ_3\rightarrow(φ_1∧φ_2),φ_4\rightarrow(φ_1∧φ_2∧φ_3),\ldots\}$$ Is it necessary that there exists $n$ such that for each $ψ$, $T⊨ψ$ implies $φ_n⊨ψ$?
Can you give me some hint ? How to generally begin such tasks ?
Remark that if for some fixed $n$, $\phi_n$ is true, then all $\phi_i$ with $i\leq n$ are true. Now, assume that all $\phi_n$ are either all true or all false.
If they are all false, then your implication is trivial.
If they are all true, then your implication is true if and only if $\Psi$ is a tautology.