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Let be sufficiently small such that.
Moreover, we choose sufficiently small such that.
we can choose sufficiently small such that (4.5).
Let be sufficiently small such that and then we get (37).
Without loss of generality, we may assume that ; then there exists a sufficiently small such that.
Since the function decreases as we can choose a sufficiently small such that and for all Moreover, (2.62).
Let be sufficiently small such that (34).
where is sufficiently small, such that.
If is sufficiently small such that (3.22).
Secondly, since, there is sufficiently small such that (2.31).
Since for there exists a sufficiently small such that (3.20).
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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