- #1

BrianMath

- 26

- 0

## 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.