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The phrase "proves impossible" is correct and usable in written English.
It is an idiomatic expression meaning "to be unable to be done or achieved". You can use this phrase in any context where you are trying to express that something is impossible to do or achieve. Example: "Attempting to please everyone proves impossible."
Exact(60)
Rapid triage proves impossible.
In rare cases this proves impossible.
But the second thorny hedge proves impossible to clear.
It's a hint that proves impossible not to take.
If that proves impossible, you'll have to abandon.
The bill could be even higher if it proves impossible to negotiate union-wide deals.
The first is that the stand-off between Cyprus and Germany proves impossible to resolve.
Matching freakishly shaped veg with their exotic names proves impossible without guidance from chef Naveen.
If that proves impossible because of British resistance, it should begin in the euro area.
Like Mr. Kagel's "Westen," this is a cross-cultural work in which assimilation proves impossible.
But it does not dictate what should happen if that goal proves impossible.
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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