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