[tsimionescu]

Is there any somewhat simple explanation of what are the limitations of the epsilon-delta definition of limits that make it non-rigorous? I've been trying to find some information about your comment, but have so far come up empty.

[JackFr]

I'm shaky on this - it's been thirty years - but I believe the Calc I epsilon delta proofs relied on the notion of an open and closed intervals on the real line, which we all intuitively understood.

The upper level Real Analysis made us bring some rigor as to what an interval on the real line actually meant going from raw points and sets to topological spaces to metric spaces, then compactness, continuity, etc. all with fun and crazy counterexamples.