- #1
avec_holl
- 15
- 0
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 {xn} = 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 < |{xn} - 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 < {xn} = 1/n < 1/N < ϵ. Since this condition is true, we have that, if n > N then 0 < |{xn} - 0| < ϵ.
To complete the proof, we need only note that since 0 < ε < {xn}, if n > N then 0 < |ε| <|{xn}| < 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!