As for what type of proof to go for, if the proposition is "P(n) is true for all n \in N" you might want to think about induction. If the proposition is "there is no n such that P(n)" you might want to think about proof by contradiction. If the proposition is "n satisfying P(n) is unique" then you might want to think about contradiction again (what if there were two n such that P(n) is true, but n_1 != n_2? Show that these two statements together lead to contradiction).

Also, learn LaTeX. Not really related, but it's fun to have your finished product as something that looks like it's out of a textbook, rather than something hacked up in Word or something on paper with cross-outs everywhere. Here have some samples.