Is this delta epsilon proof correct?

[tylerc1991]

Homework Statement

Show that Re(z) -> Re(z0) as z -> z0

The Attempt at a Solution

let epsilon > 0, choose delta = epsilon, so

|Re(z) - Re(z0)| = |(z + z')/2 - (z0 + z0')/2| (where z' and z0' are the complex conjugates)

|(z + z')/2 - (z0 + z0')/2| = |(z - z0 + z' - z0')/2| < |z - z0|/2 + |z' - z0'|/2

|z - z0|/2 + |z' - z0'|/2 = |z - z0| (since |z' - z0'| = |z - z0|)

|z - z0| < epsilon if 0 < |z - z0| < delta

Did I do this correctly? Are there any skipped steps? Thank you for your feedback!

 

[SammyS]

Your concluding line should be:

|Re(z) - Re(z₀)| < ε if 0 < |z - z₀| < δ

Otherwise, it looks good.