You sort of can't get arbitrarily close. And that's the whole reason calculus doesn't work for rationals.
Rationals fundamentally have "gaps", and correspond 1:1 with integers.
”There are also representations like
{ 0, 1, 2, 3, … | } = ω
{ 0 | 1, 1/2, 1/4, 1/8, … } = ε
where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number.”
So, ε is a gap just above zero, π-ε a gap just below π, etc.
(The difference is that, when stepping from real to real, you won’t accidentally step on a surreal in the way you’ll frequently hit irrationals when you step from rational to rational)
Maybe you have to leave those gaps unfilled.
I’ve never seen derivatives or integrals, but https://math.stackexchange.com/questions/112492/integral-of-... claims one can define them.
Is there an obvious concrete example? I've never spent proper time studying Real Analysis, and I confess I have the same intuition as OP: That you could use rationals to approximate irrationals to arbitrary precision.
What's special about real numbers is that, if you have a sequence of reals for which the distance between two consecutive elements approaches zero, then there exists a real number that's the limit of the starting sequence.
This isn't the case with rational numbers, eg. the sequence 1.4, 1.41, 1.414 ... (EDIT: these are increasing approximations of sqrt(2)) satisfies the hypothesis but there's no single rational number this sequence approaches.
This property is called completeness. The real numbers are a complete topological space, whereas the rationals aren't.
TL;DR: Every real number can be approximated by rationals. GP was being careless with language.