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The phrase "run up to" is correct and can be used in written English.
It generally means to move quickly and is used in the sense of to get closer to something. For example, "The dog ran up to the fence to bark at the passersby."
Exact(41)
Overdraft fees can run up to $40.
Fines run up to $500.
Children run up to meet him.
Newer forms run up to $500 (Tresiba, degludec).
They run up to about 2.5 miles per hour.
But the mill's load can run up to 73 megawatts.
Similar(19)
In the run-up to the announcement.
But controversy clouded the run-up to the flotation.
I haven't enjoyed the run-up to the election.
"The run-up to September is not about numbers.
The run-up to the Jets' incident was orchestrated.
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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