Let's do a little logic magic, shall we?
Note: I will assume here "$\implies$" is identical with "$\rightarrow$", and therefore that you are asserting the proposition in (1) rather than merely considering it. See discussion here for a reason why.
Let the set $S$ be the set of those elements, $x$, such that statements such as "$P(x)$" or "$Q(x)$" make sense and for such statements it makes sense to consider the propositions of the form "$P(x) \implies Q(x)$".
Consider then, since:
$$
\tag{1} \forall x \in S,P(x) \implies Q(x),
$$
$$
(1) \iff \tag{2} \forall x \in S, (\text{~}P(x) \vee Q(x)) \text{ is true.}
$$
Asserting:
$$
\tag{3} \exists y \in S, \text{~}Q(x) \text{ is true},
$$
we get,
$$
\tag{4} (\text{~}Q(x) \text{ is true}) \implies (\text{~}P(x) \text{ is true}).
$$
For the sake of brevity, I'm skipping a step here; though it is admittedly trivial.
Therefore,
$$
\tag{5} (1) \wedge (3) \implies \text{~}P(x) \text{ is true}.
$$
Q.E.D.
EDIT:
So, to answer your questions:
- You have this right.
- For most purposes, you have no problem with assumptions.
- Finally, you skipped a step between (3) and (4) in your post, which was made explicit by (2) in mine.
On a side note, for future reference, using $\LaTeX$ would probably be a better alternative to inserting mathematical notation in your posts on SE.
TIP: You can tag your equations by using "\tag{n}", where "n" denotes the number or string you want to tag your equation with.
No valid argument can prove this. Suppose $M, N, S$ and $W$ are true, and $A$ and $B$ are false. Then the three premisses are all true, but the conclusion is false.
Best Answer
It is not clear from the phrasing in your quote whether the statement "If $n$ is a real number with $n > 3$, then $n^2 > 9$" is meant to be a given premise or the statement that we're trying to prove.
If it's a premise, and we're trying to use it to prove that $n \le 3$, given the additional assumption that $n^2 \le 9$, then the argument is indeed correct: given $A \implies B$, we can deduce $\lnot A$ from $\lnot B$.
If, instead, the first statement in the quoted argument is supposed to be the theorem that we're trying to prove, then the proof is invalid. While we could indeed deduce $n > 3 \implies n^2 > 9$ from $n^2 \le 9 \implies n \le 3$ (those two statements being logically equivalent), the quoted argument contains nothing that could be used to prove $n^2 \le 9 \implies n \le 3$ in the first place, unless we incorrectly assume the statement that we're trying to prove (or something else that has not been explicitly stated).
Presumably, the book you're reading explains how such arguments given in it are to be parsed. In particular, if the arguments in the book are written in the format "Theorem to be proved: Proof of theorem," then this particular argument is invalid; if they're instead meant to be parsed as "Established premises: Assumptions. Result," then it's valid.
Unfortunately, while there exist several established conventions like this, none are quite universal enough to be safely assumed without knowing which convention the author follows. When writing proofs yourself, to avoid such confusion it's usually a good idea to clearly and explicitly indicate which statements are axioms or prior theorems, which are temporary assumptions, and which are the new theorems that you're trying to prove.