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"steadily increasing number" is a correct and usable phrase in written English.
It is often used to describe a gradual but persistent increase in the quantity of something. For example, "The steadily increasing number of people seeking medical care for mental health issues is an alarming trend."
Exact(60)
This new digital world offers a steadily increasing number of choices.
The steadily increasing number of space objects poses a considerable hazard to all kinds of spacecraft.
A steadily increasing number of Sunni militant jihadists from outside Syria have also joined the fray since mid-July.
No surprise, then, that a steadily increasing number of foreigners are pitching up here – inquisitive individuals, rather than tour groups.
Shortly after midday, long screens were set up so bystanders could not see the steadily increasing number of bodies.
The past two decades have seen a steadily increasing number of papers on bubble columns and related reactors.
The city, meanwhile, is receiving a steadily increasing number of reports of abuse and neglect — 65,856 in 2008, up from 63,434 the year before.
"On the other hand a steadily increasing number of serious crimes, including murder and rapes, are being solved and criminals brought to justice with its help.
In the past five years, a steadily increasing number of schoolchildren — as many as 15,000 a year — have come here for a course on Confucian etiquette.
On the other, a steadily increasing number of clubs, among whom the Broncos are just the most extreme example, are broke and unviable.
The steadily increasing number of examples of phenol-based redox catalysis demonstrates that the principle finds wide application in electroanalysis and related fields.
More suggestions(15)
continuously increasing number
gradually increasing number
ever increasing number
steadily increasing prevalence
steadily increasing incidence
progressively increasing number
continually increasing number
steadily increasing numbers
steady increasing number
steadily increasing ranges
steadily increasing amount
steadily growing numbers
steadily growing number
steadily increasing interest
increasingly high number
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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