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The phrase "prove their ability" is correct and usable in written English.
You can use it in any situation where you want to emphasize that someone needs to demonstrate their skill or ability in a certain area. For example, "The candidates have been asked to prove their ability in the areas of customer service and problem solving during their interview."
Exact(32)
The biotechnology companies have yet to prove their ability to translate their research into revenues, much less profits.
High-potential employees who prove their ability to quickly and efficiently acquire expertise will find themselves in great demand.
Leaders are under intense political pressure to pass a tax bill as a way to prove their ability to govern.
Among other criteria, aspiring teams will have to prove their ability to attract and develop players from their communities, as well as commercial partners.
Back in the boom, banks made loans to homeowners who did not have to prove their ability to pay, then quickly sold the loans to other companies.
To prove their ability — and to earn a certificate of completion (good for job hunting but not required) — students must mix 20 drinks in six minutes.
Similar(28)
In them, the organizations were asked to prove their abilities as educators, caregivers and marketers.
To put that another way, women have to prove their abilities, while men are assumed to have them.
They prove their abilities through their use of the applications.
Possibly, the women may be keener to prove their abilities in order to feel comfortable driving.
When employers focus only on resumes and degrees, these workers have no easy way to prove their abilities for better jobs and the "skills gap" becomes a self-fulfilling and self-defeating prophecy.
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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