Someone suggested (please see the comments below) that I post this question on hsm.stackexchange. There is a connection to the history of mathematics in this, regarding the relationship between the completeness of real numbers and the development of calculus, as well as the discovery of the incompleteness of rational numbers. Can the so-called completeness of real numbers be understood as closure under limits in the real number system?
comment： As for the mathematical analysis (your last comment), it is not that much of an achievement to prove that Q is incomplete as to prove that R is complete. In other words, you can do calculus on it and you do not need to introduce infinitesimals (which was very important milestone to achieve as it made calculus rigorous). – Stinking Bishop
@ StinkingBishop Your point is: it wasn't the development of calculus that led humanity to discover the incompleteness of rational numbers. Humanity had already realized that rational numbers were "incomplete" long before, but they didn't know that real numbers were complete. Even in the time of Pythagoras, they only became aware of the incompleteness of rational numbers; nobody had rigorously proven the completeness of real numbers. The discovery of the square root of 2 highlighted the problem but didn't solve it. The square root of 2 alone couldn't prove the completeness of real numbers.– bokabokaboka
@StinkingBishop To make calculus more rigorous, humans rigorously proved the completeness of real numbers based on the understanding that rational numbers were already incomplete. This resolved the concerns of mathematicians in the field of mathematical analysis and made calculus perfect. – bokabokaboka
Yes, that is my understanding (although the history is always messier as it happens than when we try to summarise it.) You may want to post this also on History of Science and Mathematics StackExchange. – Stinking Bishop