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Let $a,b,c$ denote complex constants. Use the definition of a limit to show that $$\lim_{z \to z_0} (az + b) = az_0 + b$$

Here is what I have done -

$$|az + b - (az_0 + b)| =$$

$$|az - az_0 + b - b| =$$

$$|a(z - z_0)| =$$

$$|a||z - z_0|$$

So for a positive number $\epsilon$,

$$|az + b - (az_0 + b)| < \epsilon \text{ whenever } |a||z - z_0| < \epsilon$$

or in other words $|az + b - (az_0 + b)| < \epsilon$ whenever $|z - z_0| < \delta$ where $\delta = \epsilon/|a|$

Have I proved the statement correctly?

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1  
Looks fine to me! –  Martin Wanvik Feb 6 '12 at 18:09
2  
Perfect, excepting one small possible issue you need to avoid in the end... You divide by $|a|$, and what happens if $|a|=0$? –  N. S. Feb 6 '12 at 18:13
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It looks like a textbook example of a $\delta$-$\epsilon$ proof to me, other than the point raised by N.S. –  robjohn Feb 6 '12 at 18:16
    
So I just make a note of that in my proof?.. where d - e/|a|, |a| != 0 –  Jim_CS Feb 6 '12 at 18:25
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You can either split the end in two cases: case 1 $|a| \neq 0$, case 2: $|a|=0$, or, pretty standard trick, observe that $|a||z - z_0| < \epsilon$ happens when $|z-z_0| < \frac{\epsilon}{|a|+1}$. –  N. S. Feb 6 '12 at 18:40

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