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We complete this lemma.
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This completes the lemma.
Together with (4.17), we complete the proof of this lemma.
Thus, we complete the proof of this lemma.
Then we complete the proof of this lemma.
Therefore, we complete the proof of this lemma.
So we complete the proof of this lemma.
By analogy with the argument taken in Lemma 2.1, we complete the proof of this lemma.
Finally we only need to fix in (4.26) to complete the proof of this lemma.
For ε 0 in Lemma 3.5, we can choose R 0 > 0 such that R 2 σ < ε 0 whenever R ≤ R 0. Then, by Lemma 3.5, one can complete the proof of this lemma immediately.
In order to complete the proof of this lemma, it suffices to show from the above inequality with respect to R ( a ) that g ( a ) < 0 for 15 100 ≤ a ≤ 1 4, where g ( a ) = 2 a ln b + ln ( 2 a ) − 2 b ln a − ln ( 2 b ) + 5 ( a − 1 4 ) − 1, b = 1 − a.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com