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From the proved part of the lemma it now follows that the function (P_{m}(t -P_{m}(0)+h_{m}(0)) has exact -P_{m zero.
From the proved above and condition 3 of Theorem 1, the inequality α n + 1 ( a ) = g ( χ ( α n ( T ) ) ) ≤ g ( χ ( α n + 1 ( T ) ) ) holds.
The estimate (4) follows directly from the proved convergence of approximating sums to the Riemann-Stieltjes integral and Lemma 1.
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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