Are there any infinitesimals in R?

[avec_holl]

Homework Statement

Prove that infinitesimals are not a subset of R.

Homework Equations

N/A

The Attempt at a Solution

Well, I had two ideas about how to prove this but I'm really not sure about either. Proof 1 was the first idea I had but I think it's probably wrong since it has to do with the limit of a sequence. Proof 2 was my other idea but I think it seems fishy so it's probably wrong. Anyway, here goes nothing . . .

Proof 1: Suppose that (∃ε)(ε ∈ R) defined by 0 < ε < 1/n ∀n ∈ N. Clearly ε satisfies the definition of an infinitesimal.

Since ε < 1/n this defines a sequence {x_n} = 1/n. We can prove that this sequence converges to zero as n becomes arbitrarily large by using the definition of convergence. Therefore,

(∀ϵ)(ϵ > 0)(∃N)(N ∈ N)(∀n)(if n > N then 0 < |{x_n} - 0| < ϵ).

Clearly, if (ϵ > 0), then the must be some (N ∈ N) such that, 1/N < ϵ. The fact that n > N implies that 0 < {x_n} = 1/n < 1/N < ϵ. Since this condition is true, we have that, if n > N then 0 < |{x_n} - 0| < ϵ.

To complete the proof, we need only note that since 0 < ε < {x_n}, if n > N then 0 < |ε| <|{x_n}| < ϵ. This implies that ε = 0 and consequently, infinitesimals are not a subset of R. Q.E.D.


Proof 2: Suppose that (∃ε)(ε ∈ R) defined by 0 < ε < 1/n ∀n ∈ N. Clearly ε satisfies the definition of an infinitesimal. Since ε ≠ 0, ε^-1 ∈ R. Because 0 < ε < 1/n ∀n ∈ N, this implies that ε^-1 > n ∀n ∈ N. However, this contradicts the Archimedean property of R and consequently infinitesimal numbers cannot be a subset of R. Q.E.D.


I hope this is the right forum for this. Please pardon any poor wording, I'm not used to formatting things for the computer. Thanks!

 

[HallsofIvy]

I think that can be done more simply as a "proof by contradiction". If an infinitesmal, $\epsilon$ were a member of R, it would have to be 0, positive or negative. It cannot be 0 by definition. If $\epsilon> 0$ then $1/\epsilon$ would be a positive real number. By the Archimedean property (which essentially is what says infinitesmals cannot be real numbers) there exist a positive integer N such that $N> 1/\epsilon$ from which $1/N< \epsilon$, a contradiction. I'll leave the last case, that $\epsilon$ is a negative real number, to you.

 

[Hurkyl]

I thought proof 1 was fine (other than neglecting the possibility of negative infinitessimals). It's the same argument -- the Archmedean principle is essentially the statement that "1/n --> 0 as n --> infinity", and his invokation of the squeeze theorem works out to essentially the same thing as your argument by contradiction.

Oh wait, nevermind -- I just noticed he didn't actually invoke the squeeze theorem, he just duplicated its proof in his argument.

Isn't his proof 2 the same thing as your argument, HoI?

 

[HallsofIvy]

Pretty much, yeah. But mine was prettier!

(There were a number of special symbols my reader could not interpret so I was not clear what he was saying.)

 

[avec_holl]

Thanks for the replies guys! I'm really glad that each approach I used will work with some modifications.

Hurkyl: Thanks for pointing out that I needed to consider negative infinitesimals. I always seem to forget some important part of a proof.

HallofIvy: Your proof by contradiction was certainly prettier than mine! Thanks for the input!

 

[Hurkyl]

avec_holl: have you worked out how to redo proof #1 in terms of the squeeze theorem? I think it's a good idea -- you were clearly thinking along those lines, and it would be a good exercise to figure out how to work efficiently with those thoughts.