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Thus it is enough to prove that for r
∈
(0
,
1)
and i
≥
0
it is
K
i
r
K
=0
.
lim
K→∞
To prove this it is enough to prove that for r
∈
(0
,
1)
,realx and a natural
i
≥
0
it is
x
i
r
x
=0
.
lim
x→∞
It is
lim
x→∞
x
i
=
,
lim
x→∞
r
x
=0
and thus according to the l'Hospital's
∞
rule it is
x
i
r
−x
(
x
i
)
(
i
)
(
r
−x
)
(
i
)
x
i
r
x
= lim
x→∞
lim
x→∞
= lim
x→∞
i
!
r
x
= lim
x→∞
=
lim
x→∞
=0
,
(
−
ln
r
)
i
r
−x
where
(
x
i
)
(
i
)
is an i-th derivation of x
i
and analogously for
(
r
−x
)
(
i
)
. This
finishes the proof.
Lemma 4.
Let us suppose a ≥
0
and b ≥
0
are natural numbers. Then it is
for each k
∈
0
,b
and
0
<p<
1
a
+
b
a
+
k
p
a
+
k
(1
p
)
b−k
=0
lim
a→∞
−
·
Proof. It is:
a
+
b
a
+
k
p
a
+
k
(1
p
)
b−k
−
=
p
a
+
k
(1
a
+
b
p
)
b−k
−
a
+
b
−
(
a
+
k
)
=
a
+
b
b
p
a
+
k
(1
p
)
b−k
−
−
k
(
a
+
b
)
b−k
p
a
+
k
(1
p
)
b−k
≤
−
·
Thus it is enough to prove that it is
(
a
+
b
)
b−k
p
a
=0
lim
a→∞
·
The proof of this assertion is similar to the proof of the assertion
K
i
r
K
=0
lim
K→∞
·
in the Lemma 3.
