(adsbygoogle = window.adsbygoogle || []).push({}); Is my proof correct for lim_(n-> infty) |z_n| = |z| ??? Complex Analysis

1. The problem statement, all variables and given/known data

Show that if lim_{n-> infty} z_n = z

then

lim_{n-> infty} |z_n| = |z|

2. Relevant equations

3. The attempt at a solution

Is this correct:

lim_{n-> infty} |z_n| = |z|

iff

Assume that the conditions hold;

lim_{n-> infty} |x_n| = |x| and lim_{n-> infty} |y_n| = |y|

According to these conditions there exist, for each positive number \epsilon, positive integers n_1 and n_2 such that:

||x_n| - |x|| < epsilon/2 whenever n > n_1

and

||y_n| - |y|| < epsilon/2 whenever n > n_2

Hence, if n_0 is the larger of the two integers n_1 and n_2

||x_n| - |x|| < epsilon/2 and ||y_n| - |y|| < epsilon/2 whenever n > n_0

Since

|(|x_n| + i|y_n|) - (|x| + i|y|) =

|(|x_n| - |x|) + i(|y_n|-|y|) <= ||x_n| - |x|| + ||y_n| - |y||

Then

||z_n| - |z|| < epislon/2 + epsilon/2 = epsilong whenver n > n_0

Thus it holds that

lim_(n-> infty) |z_n| = |z| because for every epsilon > 0 there exists N > 0 such that | |z_n| - |z|| < epsilon

I am using a proof from the text book to draw this from so just wanted to check to see if I've left anything out and if it makes sence!

Thanks

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# Homework Help: Is my proof correct for lim_(n-> infty) |z_n| = |z| ? Complex Analysis

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