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{\bf Question}

Using the $\delta-\epsilon$ definition of limit, prove that for
each $a\in {\bf C} -\{ 0\}$,
\[ \lim_{z\rightarrow\infty} az =\infty. \]
\medskip

{\bf Answer}

$\ds\lim_{z \to \infty} az=\infty$ ($a \ne 0$)

given $\epsilon>0$, there is $\delta>0$ so that if $z \in
\cup_{\delta} (\infty)-\{\infty\}$ then $az \in
\cup_{\epsilon}(\infty)$.

$$az \in \cup_{\epsilon}(\infty):\ |az|>\epsilon\ \
|z|>\ds\frac{\epsilon}{|a|}$$

Take $\delta=\ds\frac{\epsilon}{|a|}$: then if $z \in
\cup_{\delta} (\infty)-\{\infty\}$ then $|z|>\delta$ and so
$|z|>\ds\frac{\epsilon}{|a|}$ and so $|az|>\epsilon$ as desired.
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