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Exact(18)
Let with Let and be two nonnegative real sequences.
Let γ and δ be two nonnegative real numbers.
Let γ and δ be two nonnegative real numbers such that ((gamma+delta k leqgamma).
Let γ and δ be two nonnegative real numbers such that ξ ≤ γ.
Let γ and δ be two nonnegative real numbers such that ζ ≤ γ.
Let and be two nonnegative real sequences satisfying the following condition: (1.20).
Similar(42)
Let, and be three nonnegative real sequences satisfying (227).
Let ({a_{n}}), ({b_{n}}) and ({c_{n}}) be three nonnegative real sequences satisfying the following condition.
Let { a n }, { b n }, { c n } be three nonnegative real sequences satisfying the inequality a n + 1 ≤ ( 1 + c n ) a n + b n, n ≥ 1. (1.8).
Let { a n }, { b n } and { c n } be three nonnegative real sequences satisfying a n + 1 ≤ ( 1 + b n ) a n + c n, ∀ n ≥ 0. (1.8).
Let { a n }, { b n }, and { c n } be three nonnegative real sequences satisfying a n + 1 ≤ ( 1 − t n ) a n + b n + c n, ∀ n ≥ 0, where { t n } is a sequence in ( 0, 1 ).
More suggestions(2)
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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