What is Epsilon delta proof: Definition and 29 Discussions

In mathematics, the epsilon numbers are a collection of transfinite numbers whose defining property is that they are fixed points of an exponential map. Consequently, they are not reachable from 0 via a finite series of applications of the chosen exponential map and of "weaker" operations like addition and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that satisfy the equation




ε
=

ω

ε


,



{\displaystyle \varepsilon =\omega ^{\varepsilon },\,}
in which ω is the smallest infinite ordinal.
The least such ordinal is ε0 (pronounced epsilon nought or epsilon zero), which can be viewed as the "limit" obtained by transfinite recursion from a sequence of smaller limit ordinals:





ε

0


=

ω


ω


ω


















=
sup
{
ω
,

ω

ω


,

ω


ω

ω




,

ω


ω


ω

ω






,

}


{\displaystyle \varepsilon _{0}=\omega ^{\omega ^{\omega ^{\cdot ^{\cdot ^{\cdot }}}}}=\sup\{\omega ,\omega ^{\omega },\omega ^{\omega ^{\omega }},\omega ^{\omega ^{\omega ^{\omega }}},\dots \}}
Larger ordinal fixed points of the exponential map are indexed by ordinal subscripts, resulting in




ε

1


,

ε

2


,

,

ε

ω


,

ε

ω
+
1


,

,

ε


ε

0




,

,

ε


ε

1




,

,

ε


ε


ε


















,



{\displaystyle \varepsilon _{1},\varepsilon _{2},\ldots ,\varepsilon _{\omega },\varepsilon _{\omega +1},\ldots ,\varepsilon _{\varepsilon _{0}},\ldots ,\varepsilon _{\varepsilon _{1}},\ldots ,\varepsilon _{\varepsilon _{\varepsilon _{\cdot _{\cdot _{\cdot }}}}},\ldots }
. The ordinal ε0 is still countable, as is any epsilon number whose index is countable (there exist uncountable ordinals, and uncountable epsilon numbers whose index is an uncountable ordinal).
The smallest epsilon number ε0 appears in many induction proofs, because for many purposes, transfinite induction is only required up to ε0 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic).
Many larger epsilon numbers can be defined using the Veblen function.
A more general class of epsilon numbers has been identified by John Horton Conway and Donald Knuth in the surreal number system, consisting of all surreals that are fixed points of the base ω exponential map x → ωx.
Hessenberg (1906) defined gamma numbers (see additively indecomposable ordinal) to be numbers γ>0 such that α+γ=γ whenever α<γ, and delta numbers (see multiplicatively indecomposable ordinals) to be numbers δ>1 such that αδ=δ whenever 0<α<δ, and epsilon numbers to be numbers ε>2 such that αε=ε whenever 1<α<ε. His gamma numbers are those of the form ωβ, and his delta numbers are those of the form ωωβ.

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  1. H

    I ##\epsilon - \delta## proof and algebraic proof of limits

    It occurred to me that I should ask this to people who passed the stage in which I’m right now, being unable to find anyone in my milieu (maybe because people around me have expertise in other fields than mathematics) I reckoned to come here. Let’s see this sequence: ## s_n =...
  2. I

    Epsilon delta proof of the square root function

    Let ##\varepsilon > 0## be arbitrary. Now define ##\delta = \text{min}\{\frac{a}{2}, \varepsilon \sqrt{a}\}##. Now since ##a>0##, we can deduce that ##\delta > 0##. Now assume the following $$ 0< |x-a| < \delta $$ From this, it follows that ##0 < |x-a| < \frac{a}{2} ## and ##0 < |x-a| <...
  3. M

    Ε-δ proof: lim x->a f(x) = lim h->0 f(a + h)

    This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate. Homework Statement Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h) Homework EquationsThe Attempt at a Solution By the...
  4. J

    Help with Epsilon Delta Proof of Multivariable Limit

    Homework Statement Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus can't ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1). Homework Equations...
  5. Alpharup

    I Why non-standard analysis is not used?

    In analysis, we do encounter tougher epsilon-delta proofs instead of more intuitive algebraic methods( those involving infintesimals). I have read that there is branch where infintesimals are rigorized like epsilon-deltas. My question is why people don't use that? Also, is it logically sound...
  6. TheSodesa

    A multivariable limit problem (epsilon-delta -proof)

    Homework Statement Find the limit \lim_{(x,y)\to(2,2)}\frac{x^3-y^3}{x-y} Homework Equations \epsilon - \delta, baby: If the limit L exists, \forall \: \epsilon \: \exists \: \delta: 0 < \sqrt{(x-a)^2+(y-b)^2} < \delta \rightarrow |f(x,y)-L| < \epsilon The Attempt at a Solution By...
  7. AutumnWater

    Epsilon Delta proof of a 2variable limit. Is my proof valid?

    Homework Statement Use the epsilon delta definition to show that lim(x,y) -> (0,0) (x*y^3)/(x^2 + 2y^2) = 0 Homework Equations sqrt(x^2) = |x| <= sqrt(x^2+y^2) ==> |x|/sqrt(x^2+y^2) <= 1 ==> |x|/(x^2+2y^2)? The Attempt at a Solution This limit is true IFF for all values of epsilon > 0, there...
  8. F

    Where is the mistake in this epsilon delta proof?

    The question asks to proof that the limit given in incorrect by contradiction. I tried working using the estimation method and the weird thing is that I completed the proof and found that the supposedly "incorrect" limit gave a correct answer although it was supposed to give me a contradiction...
  9. Dethrone

    MHB Epsilon Delta Proof for 1/x

    $$\lim_{{x}\to{2}}\frac{1}{x}=\frac{1}{2}$$ Here is what I have so far: For all $\delta >0$, there exists an $x$ such that $0<|x-2|<\delta $, $|\frac{1}{x}-\frac{1}{2}<\epsilon$ Cut to the chase: $$\frac{|x-2|}{|2x|}<\epsilon$$ I need to bound $\frac{1}{|2x|}$ somehow, and represent it with...
  10. Dethrone

    MHB Epsilon Delta Proof Piecewise function

    https://answers.yahoo.com/question/index?qid=20130915100124AAK4JAQ I do not understand how they got: "1 = |(1 plus d/2 - L) - (d/2 - L)| <= |1 plus d/2 - L| plus |d/2 - L| < 1/4 plus 1/4 = 1/2, " Shouldn't it be $|(1+ \frac{\delta}{2} -L) + (\frac{\delta}{2} -L)|$, not $|(1+ \frac{\delta}{2}...
  11. D

    Epsilon Delta Proof of a Limit

    Hey there, I'm new to this forum. Today I thought I would brush up on my calculus. I would just like to know if my method is correct. Is there an easier way to prove this ? By the way, it's my first time using LaTeX, so bear with me. I am trying to prove the following : \lim_{x\rightarrow...
  12. LeibnizIsBetter

    MHB Epsilon delta proof of a two-variable limit using inequalities

    I seem to be having trouble with multivariable epsilon-delta limit proofs. I don't have a very good intuition for how \epsilon relates to \delta. For example: Prove \lim_{(x,y) \to (0,0)}\frac{2xy^2}{x^2+y^2} = 0 There are probably many ways to do this, but my teacher does it a certain way...
  13. Seydlitz

    Variation of Epsilon Delta Proof

    Homework Statement Prove that if ##\left |x-x_{0} \right | < \frac{\varepsilon }{2}## and ##\left |y-y_{0} \right | < \frac{\varepsilon }{2}## then ##|(x+y)-(x_0+y_0)| < \varepsilon ## and ##|(x-y)-(x_0-y_0)| < \varepsilon ##Homework Equations Postulate and proof with real numbers as well...
  14. D

    Need help with epsilon delta proof of f(x)=x^4+(1/x) as x goes to 1

    Homework Statement Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ. f(x)=x^{4}+\frac{1}{x}, a=1. Homework Equations I claim that \lim\limits_{x\rightarrow 1}x^{4}+\frac{1}{x}=2. The...
  15. D

    Epsilon delta proof that x^4 goes to a^4 as x goes to a

    Homework Statement Determine the limit l for a given a and prove that it is the limit by showing how to find δ such that |f(x)-l|<ε for all x satisfying 0<|x-a|<δ. f(x)=x^{2}, arbitrary a.Homework Equations I will incorporate the triangle inequality in this proof.The Attempt at a Solution We...
  16. F

    Solving Epsilon Delta Proof: lim 3 as x->6 & lim -1 as x->2

    Homework Statement lim 3 as x->6 lim -1 as x->2 Homework Equations In the first weeks of a calculus class and doing these epsilon delta proofs. As i am looking at two of the problems i have been assigned: Lim 3 as x->6 Lim -1 as x->2 The Attempt at a Solution...
  17. A

    Solving Epsilon-Delta Proof: Homework Statement

    Homework Statement I just want to make sure I include all the steps in doing this: lim (6x-7) = 11 x->3 Homework Equations The Attempt at a Solution given ε>0, we need to find a δ>0, such that 0< lx-3l < δ then 0 < l (6x-7)-11 l < ε To prove this I need to make 0 < l...
  18. K

    Solving Epsilon Delta Proof: |x^2 - 9| < ε

    Homework Statement if |x-3| < ε/7 and 0 < x ≤ 7 prove that |x^2 - 9| < ε Homework Equations The Attempt at a Solution So ths is what I did so far. |x+3|*|x-3| < ε (factored out the |x^2 - 9|) |x+3|*|x-3| < |x+3|* ε/7 < ε (used the fact that |x-3| < ε/7) |x+3|* ε/7 *7 <...
  19. J

    Beginner's Epsilon Delta Proof: Help needed

    Homework Statement This is my first delt/epsilon proof ever, so please understand if I seem ignorant. e=epsilon d = delta Let f(x) = 1/x for x>0 If e is any positive quantity, find a positive number d, which is such that: if 0 < |x-2| < d, then |f(x) - 1/2| < e Homework...
  20. D

    Help on epsilon delta proof of discontinuity

    Homework Statement Prove the function f(x)= { 4 if x=0; x^2 otherwise is discontinuous at 0 using epsilon delta. Homework Equations definiton of discontinuity in this case: there exists an e>0 such that for all d>0 if |x-0|<d, |x^2-4|>e The Attempt at a Solution I'm confused...
  21. L

    Help With Epsilon Delta Proof

    I have a problem on a take-home test, so I can't ask about the specific problem. So this is just going to be a general, how do I put stuff together problem. I have a function of x and y that maps R2 into R1. The limit as (x,y)->(0,0) is zero, and I've worked through the various paths already...
  22. E

    How is the epsilon delta proof an actual proof?

    I am a first year freshman at UC Berkeley, in Math 1A. We learned the delta-epsilon proof for proving the limit of functions. I won't go through a whole proof or anything, but the general idea is you have a delta that is less than |x-a| (and greater than zero) and an epsilon less than |f(x)-L|...
  23. H

    Help Doing an Epsilon Delta Proof

    Homework Statement given a function defined by f(x,y) {= |xy|^a /(x^2+y^2-xy), if (x,y) cannot be (0,0) and = 0, if (x,y) = (0,0) Find all values of the real number a such that f is continuous everywhere e= epsilon d= delta In order to prove this, I know I need to do an...
  24. R

    Solved Epsilon-Delta Proof: Is it Coherent?

    [SOLVED] Epsilon Delta Proof Does this limit proof make total sense? Given : "Show that \lim_{x \rightarrow 2} x^{2} = 4." My attempt at it :0<|x^{2}-4|<\epsilon which can also be written as 0<|(x-2)(x+2)|<\epsilon. 0<|x-2|<\delta where \delta > 0. It appears that \delta = \frac...
  25. D

    Proof ε-δ: Epsilon Delta Proof of lim x->a of ((3x²-3a²)/(x-a)) = 6a

    I’m going to say from the beginning that I need to hand this problem in. I'm not looking for the answer, I think I already have it, just want a critical eye. I need someone to look over this problem and tell me if it's good. Not just if it's right but if it's perfect. I always get the...
  26. H

    Proving Lim x = -2, x^2+3x+7 = 5 | Epsilon Delta Proof Help"

    I'm supposed to prove that lim x -> -2, x^2+3x+7 = 5 Here's what I have: |x^2 +3x+7 – 5| < ε |x+2| < δ |x^2+3x+2| -> |(x+2)(x+1)| < ε whenever |x+2| < δ |x+1||x+2| < δ |x+1| |x+1||x+2| < δ|x+1| < ε ε / |x+1| > δ , as x -> -2, |x+1| -> 3, therefore: ε / -1 > δ But...
  27. G

    Epsilon delta proof, 3-space help

    I am trying to show that a certain function, f(x) has a limit that approaches 1. Does anyone have any sites i can look at for epsilon delta proof for 3-space? I've saw the ones for two space, but they aren't really helping me out in this pickle.. thanks.
  28. H

    Epsilon Delta Proof, need clarification

    Prove that lim_{x \rightarrow c} \ \ \frac{1}{x}=\frac{1}{c} \ ,c\neq0 Proof We must find \delta such that: 1. 0<|x-c|<\delta \ \Rightarrow \ | \ \frac{1}{x}=\frac{1}{c}|<\epsilon Now, 2. | \ \frac{1}{x}=\frac{1}{c}|=| \ \frac{c-x}{xc}|= \frac{1}{|x|} \cdot...
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