[zeroonetwothree]

Complete just means the limit of every sequence is part of the set. So there’s no way to “escape” merely by going to infinity. Rational numbers do not have this property.

How to construct the real numbers as a set with that property (and the other usual properties) formally and rigorously took quite a long time to figure out.

[lupire]

Critically, "Complete" also means that the supposed limit necessarily exists.

[freehorse]

... for "Cauchy sequences", which are basically sequences whose terms become "closer and closer together".

You can still have sequences with no limits (a_n:=n, going to infinity, where all successive terns differ by 1 and which does not have a limit in the usual metric), as well as sequences with multiple limit points (in which case, subsequences can be considered).

Btw this is "Cauchy completeness", so it is a bit different (but equivalent) way to approach the construction of the real numbers from Dedekind's, but it is also one that can apply to more general metric spaces.