Objection!

One idea I'd like (esp. in a text, but in a class it's also okay) is
to have an -objection- every time I gloss over a detail.

In the text, this can work by having a little "objection" box after
any statement/proof that's not quite correct;
in class, this can consist of letting students say "objection!" every
time I gloss over a point (indeed, soliciting it).
[Note that in Russia, the tradition is:
 if you make a mistaken statement at the board,
 someone will raise their hand or call out "counterexample" or
 "mistake" -- and not state what the counter-example is:
 let the speaker correct it.]

Further, for each objection, have an IOU: a place where you correct
the objection.
Esp. good is if you have forward/backwards references, and each
objection/IOU is numbered.
EG:
a proof of the Fund.Thm. -before- having defined continuity rigorously.
Thm: (FTC)
If $f$ is continuous, then $F(x) := F_a(x) := \int_a^x f$ is differentiable,
and $F'(x)=f(x)$.
Pf: Since $f$ is continuous, $f(x+h) \approx f(x)$
so $F(x+h)-F(x)=\int_x^{x+h} f \approx h \cdot f(x)$ (rectangle)
so (write definition of derivative) $F'(x) = f(x)$.
Objection 17: $\approx$ is vague: we need to be precise and show why
in the limit it becomes an equality.
We need to say -how close- the approx is: that's analysis.
(see p.235 for IOU)

later:
IOU 17: (see p.123 for the objection this fixes)
Thm: (good proof of FTC): eventually within $\epsilon$ for all
$\epsilon$, etc.

WTF is the -point- of this?
1: encourage students to become -careful- readers
2: be honest, but not too soon:
  if you're -completely- rigorous right away, it's unreadably formal,
  esp. for immature (inexperienced) students

----------------------------
Note that this is a key way that -reference- books are different from
-teaching- books: teaching books generally assume -less- maturity, so
they don't do everything completely correctly the first time.
[OTOH, I think that in most other ways, reference/teaching books
 should be very similar: same amount of motivation, same examples,
 etc.;
 The other point is that inexperienced folk won't understand examples
 from other areas: Spec is not the first topological space they see.]
