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Now we prove the assertion (i).
Assertion (i) of Theorem 1.1 follows directly.
By Lemma 2.1, the assertion (i) holds.
Assertion (i) is easy to check.
We first show the assertion (i).
Assertion (ii) is a global version of assertion (i).
end{aligned} (2.1) This proves assertion (i) of the theorem.
Then the desired assertion (i) follows from Lemma 1.
Proof: We give the proof for assertion (i).
This implies the analogue of Lemma 1 and the main assertion (i) ⇒ (ii) of Theorem 12.
By assertion (i) we can further get that (|u_{n}|_{v}>r).
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com