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

• Unassuming
In summary, the statement is asking to prove that the limit of z_n equals z_0 if and only if the conjugate of z_n equals the conjugate of z_0 as n goes to infinity. This can be shown by using properties of complex conjugates and rewriting the statement as a proof about limits.
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.

## 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.

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.

Dick said:
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.

## 1. What is the meaning of "conjugate" in this statement?

"Conjugate" refers to the complex conjugate of a complex number, which involves changing the sign of the imaginary part. For example, the complex conjugate of 3+4i is 3-4i.

## 2. How do you know if a complex number is convergent?

A complex number is convergent if its real and imaginary parts both approach a finite limit as the independent variable approaches a certain value. In other words, the real and imaginary parts of the complex number must become closer and closer to a constant value as the independent variable gets closer to a specific value.

## 3. Does the statement apply to all complex numbers?

Yes, the statement "If a complex converges, then it's conjugate converges" applies to all complex numbers, as long as the complex number in question is convergent.

## 4. Can a complex number converge without its conjugate converging?

No, if a complex number is convergent, then its conjugate must also be convergent. This is because the real and imaginary parts of a complex number are dependent on each other, and if one part is converging, the other part must also be converging in order for the entire complex number to converge.

## 5. What is the significance of this statement in mathematics?

This statement is important in complex analysis, as it helps to determine whether a function is analytic (differentiable) at a point. If a complex number and its conjugate both converge, then the function is analytic at that point. This statement also has applications in the study of complex series and their convergence.

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