At least in the Netherlands, construction of the reals (for example from rational Cauchy sequences) is standard 1st semester stuff. Understanding reals is required or provides a good source of examples for virtually all mathematics courses, so I can't imagine how some universities teach mathematics without it.

Actually one of the ways to prove that requires the fact that R completes Q. The concept of completeness was introduced by Cauchy in the 19th century and it might not have been completely popularised in 1869.