If a complex converges, then it's conjugate converges.

[Unassuming]

Homework Statement

Prove that z_n -> z_0 if and only if ~(z_n) -> ~(z_0) as n goes to infiinity.

~(z_n) is the conjugate of z_n.

Homework Equations

The Attempt at a Solution

|~(z_n) - ~(z_0) | = | ~(z_n) + ~(-z_0)| <=

|~(z_n)| + |~(-z_0) | = |z_n| + |z_0| <=

and I can't come up with much else. It's about the same for the other direction as well.

 

[Dick]

Using a '*' for complex conjugation is more usual than '~'. |z*|=|z|.
|z_n*-z_0*|=|(z_n-z_0)*|=|z_n-z_0|. Use stuff like that. Now write it in the form of a proof about limits.

 

[Unassuming]

Using a '*' for complex conjugation is more usual than '~'.

* looks better after seeing it. I just couldn't think of a way the first time around.