1. ## Convergence

Hello all,

I have the following problem:

Let $\displaystyle z_{n}$ be a complex converging sequence.
We want to show that $\displaystyle \mid z_{n} \mid$ also converges.

I have done the following:

We want to show that $\displaystyle \mid z_{n} \mid$ converges which means that we have to show that for all $\displaystyle \epsilon >0$ there exists $\displaystyle N\leq n$ such that $\displaystyle \vert \vert z_{n}\vert-L \vert <\epsilon$- We have:

$\displaystyle \vert \vert z_{n}\vert-L \vert = \vert \vert z_{n}\vert- \vert L \vert \leq \vert z_{n} -L \vert < \epsilon$

The last inequality comes from the fact that z_{n} converges.

Is the above proof correct? I think that I am assuming that the limit of $\displaystyle \mid z_{n} \mid$ and $\displaystyle z_{n}$ are the same??!!

thanks

2. ## Re: Convergence

$L < 0 \Rightarrow ||z_n| - L| \neq ||z_n| - |L||$

3. ## Re: Convergence

How about rather than "assuming that the limit of $\displaystyle z_n$ and $\displaystyle |z_n|$ and are the same", taking the limit of $\displaystyle z_n$, which you know exists, and the limit of $\displaystyle |z_n|$ to be |L|.

4. ## Re: Convergence

Originally Posted by surjective
I have the following problem:
Let $\displaystyle z_{n}$ be a complex converging sequence.
We want to show that $\displaystyle \mid z_{n} \mid$ also converges.
Actually the problem should be: If $z_n\to L$ then $|z_n|\to |L|$.

lemma: $||z|-|w||\le|z-w|$
proof:
\begin{align*}|z|&=|z-w+w|\\&\le|z-w|+|w|\\|z|-|w|&\le|z-w|\\\\|w|&=|w-z+z|\\&\le|w-z|+|w|\\|w|-|z|&\le|w-z| \end{align*}

Because $|z-w|=|w-z|$
\begin{align*}-(|z-w|)&\le (|z|-|w|)\le |z-w|\\||z|-|w||&\le |z-w|\end{align*}