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Lemma.
for all constant values of .
Proof. Let
be any positive number. Then we get:
as soon as
. Here,
means the ceiling function; has been estimated downwards by taking only one of the all positive terms of the series expansion
Proof. Since , we obtain by using the lemma the result
Corollary 1.

Proof. According to the lemma we get
Corollary 2.

Proof. Change in the lemma to .
Corollary 3.
(Cf. limit of nth root of n.)
Proof. By corollary 2, we can write:
as
(see also theorem 2 in limit rules of functions).
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