Is the flow of my logic flawed?

No, he knows it's a limit. He's just got a messed up understanding of limits. He says they "approach" as if approaching is an infinite process of almost equaling.

It's hard to explain his views. Here's a short excerpt:
 [...] Me: In short, given that .999... = lim{.9, .99, .999, ...}, and as you just admitted, lim{.9, .99, .999, ...} = 1, we can use the transitive property of equalities to say .999... = lim{.9, .99, .999, ...} = 1, therefore .999... = 1. Him: In the sense of a limit, it does equal one. However, it is nonetheless a limit. Me: What other sense can a limit have, other than that of a limit? Him: No matter what view you take this problem from, it will always have to have a limit. Whenever you deal with infinite numbers, you have to have a limit meaning that the answer you get will only ever approach.

 Then he needs to define what he means by limit, because the word limit in this context for math means: Limx →pf(x)=L if and only if for any given ε > 0 there exists a δ > 0 such that 0 < | x − p | < δ implies | f(x) − L | < ε Notice the limit is equal to L with the "=" symbol by definition, nowhere is the "≈" in this definition.

I've pretty much tried that:
 Me: It is incorrect to say that a limit approaches but does not equal. The value a limit approaches defines the value of the limit itself, thus approaches implies equals. Him: You are correct in saying that mathematical notation and pragmatism enforce the use of "equals" when evaluating limits. However, when you deal with infinity, it is a touchy subject. You can never actually plug in infinity, so the limit you take will always only be moving toward the value in theory.
Is there any good reference that explicitly states that limits equal a value? Wikipedia won't work.

 Any calculus textbook should state the formal definition of a limit I’ve given you.
 And infinity is not a touchy subject. It has a precise mathematical definition in this context: Limx →∞ f(x)=L if and only if for every ε>0 there exists N > 0 such that |f(x) – L| < ε whenever x > N. Limits have nothing to do with plugging stuff in. A limit is a precise mathematical concept. The words “plug in” don’t appear anywhere in the definition. Your friend is trying to redefine terms to make himself right.

 Quote by TylerH He's stuck on "it will approach, but never equal" which forces me to have to try to come up with some contrived real number theory method of proving I'm right. I know that his interpretation of a limit as an infinite process of approaching is wrong, but I have no way to argue it. Suggestions?
I don't think it's contrived at all! In fact it's my favorite way of saying that .999...=1. You only need a basic fact about the real numbers and waam! It's done.

 Quote by TylerH No, he knows it's a limit. He's just got a messed up understanding of limits. He says they "approach" as if approaching is an infinite process of almost equaling. It's hard to explain his views. Here's a short excerpt:

And if he refuses to agree with a formal definition, then you are wasting your time. He is not talking about the same mathematics you are talking about.

Mentor
 Quote by TylerH Mark44: I completely see your argument. The problem is that when I argued 1-.999...=0, my friend said 1-.999...=1/∞, which is right
No, it isn't. 1/∞ is not a number, so it can't be the answer to a subtraction problem. The correct answer to 1 - .999... is 0.
 Quote by TylerH , but he refuses to admit that limits are the constant value that is approached(0, in this case), rather than the never ending sequence. He's stuck on "it will approach, but never equal" which forces me to have to try to come up with some contrived real number theory method of proving I'm right. I know that his interpretation of a limit as an infinite process of approaching is wrong, but I have no way to argue it. Suggestions?

 Recognitions: Gold Member Homework Help Science Advisor Perhaps you might ask him what he "means" by a real number. For example, how do you construct them? By Dedekind cuts, perhaps, or as equivalence classes of bounded, increasing sequences?

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