Precisely as Thimoteus says.
If you're having trouble with how to
Argue that [for] any statement in propositional logic [you] may ...
You might try remembering what a well formed formula (wwf) is, and how it's defined, before trying to answer to your problem. How do you decide whether any arbitrary formula is
statement in propositional logic?
You might do that so automatically, that you don't even consciously think about it, but consider, how would you program a computer to check a (finite) string of symbols and output true or false when it's syntactical or not?
IIRC, here's an almost identical question that should help you immensely
How do you show that any statement in propositional logic can expressed by the sheffer stroke?
It's enough to show a basic sets of operators like {¬, ∨} are expressible ({¬, ∧}, and {¬, →} are probably equally good sets to pick too. They're all probably just different operator bases for the same function space, a sort of vector space, over the wffs), and recalling that all other operators are defineable in terms of them.
Truth tables might be enough
| A | B | A↓B |
|---|---|---|
| T | T | ⊥ |
| T | ⊥ | T |
| ⊥ | T | T |
| ⊥ | ⊥ | T |
| A | A ↓ A | ¬ A |
|---|---|---|
| T | ⊥ | ⊥ |
| ⊥ | T | T |
| A | B | A → B | A↓B | (A↓B)↓(A↓B) |
|---|---|---|---|---|
| T | T | T | ⊥ | T |
| T | ⊥ | ⊥ | T | ⊥ |
| ⊥ | T | T | T | ⊥ |
| ⊥ | ⊥ | T | T | ⊥ |
or you might prove some suitably cunning theorem showings they're equivalent (e.g show ⊨(A ↓ A) ↔ ¬ A somehow).
Then you show by induction (on wffs) that for every well formed formula that can be expressed with just those, there's one (and only one!) logically equivalent one that exists, just in sheffer notation. That's not exactly a trivial argument to make. You have to show there exists one well formed sheffer expression for each normal wff, and that's they're logically equivalent, all in one big induction argument.
Very roughly, it's a bit like showing of statements in arithmetic that's those true in decimal is also true in base 2, or octal, or that whatever you can say in English, there's an equivalent sort of expression in French, though there's some very important differences.
If you need help with that, it's probably best to make it a separate post.