Prove that $\vdash A\to \lnot\lnot A$
By Deduction Rule we know that it is sufficient to show that ${A}\vdash \lnot\lnot A$
I am also familiar with the formula: $\lnot A \vdash (A\to B)$. So if I set $B:= \lnot\lnot A$ I get: $$\lnot A\to (A\to \lnot\lnot A)$$
I could use $MP$ but I assumed before $A$ and not $\lnot A$.
I'd be glad for help
Let's rewrite $\neg A$ as $A \to \bot$. Then $\neg \neg A$ is $(A \to \bot)\to \bot$. You eliminate the implication, and get $A, A \to \bot \vdash \bot$. You need to prove false, and you have both $A$ and that $A$ implies false.