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**Equivalent Defintions of **
It is well-known that for all complex numbers *z*,

What follows below is a direct proof of this fact in the case where *z* is a
positive real number. The proof is elementary in that it does not depend on
limits superior and inferior, but instead on the Pinching Theorem.
**Lemma 198**

Suppose that and are positive integers and .Then

**Proof:** The lemma is clearly true if

since the righthand side expression is never negative. Therefore, suppose that
In this case the proof is by induction on *k* along with the observation that
for , . **QED**
It follows directly from the Binomial Theorem that for and *x* > 0

so on the one hand we have
while by applying the Lemma we have
so that
Since it is readily established that
exists for positive by using comparison to a geometric series,
we see that
has the same limit as .

*Eric S Key*

*10/24/2000*