Direct Proofs: Are They Just Introductions?

[Benn]

Hey guys,

I'm in a proof class right now. We've covered direct proofs and moved on, but I'm still curious about them. Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs? In class, we only ever cover proofs such as "if n $\equiv$ 1 (mod 2), then n² $\equiv$ 1 (mod 8)." and the like.

Sorry, I can't get the tex to work out... aha, just got it working, nevermind

 

[jgens]

Is there any important theorem that has even been derived using a direct proof (assume p to show q) or are they mainly just used to introduce proofs?

Plenty of important theorems are proved using methods of direct proof. I presume that you are familiar with the fundamental theorems of calculus. The standard proofs of these results are done via direct proof. See here: http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#Proof_of_the_first_part