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Providing an explicit, systematic, and fruitful definition of the real numbers constitutes a major step towards completing the arithmetization of analysis.
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That relationship is already fruitful.
Being in my mid-50s has felt unmoored, which is a very fruitful place to be, partly because it's liminal — middle life is a transition moment, by definition, and so is insomnia: You're precariously balanced between two worlds.
It seems likely that this is a more fruitful – if imperfect – model than one which follows a client group model which would introduce further fissures around definition and transition.
Fruitful scholarship followed.
The misunderstanding was fruitful.
Anxiety is fruitful.
The decision proved fruitful.
A united, fruitful effort.
It was fruitful.
'The fruitful tree bends.' ".
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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