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"increase relentlessly" is correct and can be used in written English.
You can use this phrase when discussing something that increases continuously and without any signs of slowing down or stopping. For example, "Global temperatures have been increasing relentlessly since the industrial revolution."
Exact(2)
"We need to consume less … And yet, every decade, global consumption continues to increase relentlessly".
More broadly, however, ads' thirst for bandwidth seems destined to increase relentlessly, whether or not bandwidth itself increases quickly enough to meet that need.
Similar(58)
The pressure to improve a school's Sats results and league-table position increases relentlessly year on year.
They also made many Swiss feel prickly and victimised.The pressure to erode Switzerland's bank-secrecy laws has increased relentlessly.
Processing fluency is itself the product of repeated experience, and it increases relentlessly with the number of times we have the experience.
Are governments and the world's major oil, coal and gas companies – some of the most influential corporations on Earth – really going to decide to leave the money in the ground, as demand for energy increases relentlessly?
"There is some concern within the industry that demand is increasing relentlessly - particularly in countries like China, where the standard of living is increasing and people are getting a taste for some of these confectionaries," Prof Hadley said.
The prevalence of childhood overweight is increasing relentlessly (1– 2).
The flood of scientific data is increasing relentlessly, and this offers many novel opportunities.
The demand for intensive care has increased relentlessly over the past 30 years.
Type 2 diabetes mellitus (hereafter referred to as diabetes) has been increasing relentlessly since the second half of the 20th century [ 1- 3].
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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