TITLE: The role of 'arbitrary' in proofs
QUESTION [12 upvotes]: Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..."
My question is, why does considering an arbitrary element in a proof imply that the proven result applies to every element in the set? Although I think I have an vague about this, I am still interested in seeing what others have to say in relation to this idea. Perhaps, if possible, answer the question in the manner that you would if a student of yours posed this question to you.
Thank you!

REPLY [2 votes]: Let's say we have an argument. You say "for all x in the set S, A (x) is true". I say "I don't believe it. I'm sure there is an x in the S where A (x) is not true". You say "well, tell me an x and I'll show you A (x) is true". I say "I can't think of any x right now, so just assume any x". You say "Ok then, we will assume that x is any element of the set S, and I'll prove that A (x) is true. And since we don't make any assumption about that x, the same proof will work for any x in the set, so S (x) is true for all x".