Is My Epsilon-Delta Proof Correct?

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In summary, using the Epsilon-Delta Definition of a limit, it has been shown that \lim_{z\to z_0} \overline{z} = \overline{z_0} where \overline{z} is the conjugate of z. This proof follows the logic of choosing \delta = \varepsilon and showing that for any z such that 0 < | z - z0 | < δ, then it follows (by algebra, etc.) that \left|\bar{z}-\bar{z}_0\right|<\epsilon\,.
  • #1
BrianMath
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Homework Statement


Using the Epsilon-Delta Definition of a limit, show that

[tex]\lim_{z\to z_0} \overline{z} = \overline{z_0}[/tex]

where [itex]\overline{z}[/itex] is the conjugate of z.

Homework Equations


[tex]|\overline{z}|=|z|[/tex]
[tex]\overline{z_1}-\overline{z_2} = \overline{z_1-z_2}[/tex]


The Attempt at a Solution


[tex]\forall \varepsilon > 0, \exists \delta > 0: 0 < |z-z_0| < \delta \implies |\overline{z} - \overline{z_0}| < \varepsilon[/tex]
[tex]|\overline{z}-\overline{z_0}| = |\overline{z-z_0}| = |z-z_0|[/tex]
Choose [itex]\delta = \varepsilon[/itex]

[tex]|\overline{z-z_0}| = |z-z_0| < \delta = \varepsilon[/tex]


I'm currently teaching myself Complex Variables using Churchill and Brown's text, but do not have access to any instructors to comment on my work (summer before university). This is the first time that I'm seeing [itex]\varepsilon-\delta[/itex] proofs, so I want to make sure that my proof is correct.
 
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  • #2
You've got the main part sort of backwards. What you've shown is what I call the scratch work that you do on the side before writing up the formal proof. It's how you decide what relationship δ should have to ε so that when z is within the 'δ - neighborhood' of z0, then your function of z, in this case [itex]\bar{z}[/itex], is within the 'ε - neighborhood' of the limit L, in this case [itex]\bar{z}_0[/itex].

Yes, you start with ε > 0 as you did. Then you set δ, in this case you said δ = ε.

Next show that for any z such that 0 < | z - z0 | < δ, then it follows (by algebra, etc.) that [itex]\left|\bar{z}-\bar{z}_0\right|<\epsilon\,.[/itex]
 
  • #3
Okay, thank you. I was wondering if I had the right order of everything.
So is this right, now?

Let [itex]\varepsilon > 0[/itex] and choose [itex]\delta = \varepsilon[/itex]. Then
[tex]0 < |z-z_0| < \delta \implies |\overline{z}-\overline{z_0}| = |\overline{z-z_0}| = |z-z_0| < \delta = \varepsilon[/tex]
 
Last edited:
  • #4
BrianMath said:
Okay, thank you. I was wondering if I had the right order of everything.
So is this right, now?

Let [itex]\varepsilon > 0[/itex] and choose [itex]\delta = \varepsilon[/itex]. Then
[tex]|z-z_0|<\delta \implies |\overline{z}-\overline{z_0}| = |\overline{z-z_0}| = |z-z_0| < \delta = \varepsilon[/tex]
That's much better.

Probably should say [itex]0<|z-z_0|<\delta \implies |\overline{z}-\overline{z_0}\,| = \dots[/itex]

You don't want (or need) z = z0.
 
  • #5
SammyS said:
That's much better.

Probably should say [itex]0<|z-z_0|<\delta \implies |\overline{z}-\overline{z_0}\,| = \dots[/itex]

You don't want (or need) z = z0.

Oops, oh yeah, I did forget the greater than 0 part. :biggrin:

Thank you very much for your help.
 

Related to Is My Epsilon-Delta Proof Correct?

1. How do I know if my epsilon-delta proof is correct?

To ensure the correctness of your epsilon-delta proof, you should follow the standard format and steps for constructing such proofs. This includes starting with stating the given limit, choosing an arbitrary epsilon value, and finding a corresponding delta value. Then, you should clearly explain how you arrived at your chosen delta value and how it relates to the given epsilon. Finally, you should use algebraic manipulation to show that the chosen delta indeed satisfies the definition of the limit. It is also helpful to double-check your calculations and reasoning for any errors.

2. What are the common mistakes to avoid in an epsilon-delta proof?

Some common mistakes to avoid in an epsilon-delta proof include using a specific value for epsilon instead of an arbitrary one, not clearly stating the given limit, not showing the relationship between delta and epsilon, and making incorrect algebraic manipulations. It is also important to carefully consider the domain of the function and ensure that your chosen delta works for all values within that domain.

3. Is it necessary to use the epsilon-delta definition when proving a limit?

While it is not always necessary to use the epsilon-delta definition when proving a limit, it is often considered the most rigorous and comprehensive method. It allows for a clear understanding of how the limit behaves as the input approaches a certain value and can be used to prove limits for all types of functions, including discontinuous ones.

4. Can I use a graph to prove my epsilon-delta proof is correct?

Using a graph can be helpful in visually understanding the behavior of a function and its limit. However, a graph alone is not sufficient to prove an epsilon-delta proof is correct. It is important to provide a mathematical explanation and algebraic manipulation to show how the chosen delta satisfies the given epsilon.

5. How can I improve my skills in constructing epsilon-delta proofs?

The best way to improve your skills in constructing epsilon-delta proofs is to practice regularly. Start with simpler examples and work your way up to more complex ones. It can also be helpful to seek feedback from peers or a mentor, and to study and analyze well-written proofs. Additionally, familiarizing yourself with common functions and their limits can aid in understanding the steps and reasoning behind constructing an epsilon-delta proof.

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