Constructing the reals is a worthy topic;
worthy of an appendix, probably, but worthy nonetheless.

I think one should go the whole hog, and construct:
N -> Z -> Q -> R -> C

Note that the key steps are:
- peano's axioms and the natural numbers
  (so many bad mneumonics come to mind: peano/piano)
- Q -> R

The rest is easy algebra.
Do it, and a lil' bit of group/ring/field theory.
N -> Z
  is usual semigroup -> group
Z -> Q
  is "field of fractions",
  and is a -great- way to talk about equivalence classes
  (EG, (2,3) ~ (4,6) is just 2/3 = 4/6;
   note that this is quite similar to construction of projective line;
   indeed, I think $\bP\bZ = \bQ \cup \set{\infty}$)
R -> C
  is everyone's favorite field extension
  (talk about Q -> Q(sqrt(2)) too, and field automorphisms)
...so these allow you to inject some fun fun algebra into calculus class!

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construction of N (peano) lets you think about induction more
precisely: the natural numbers are exactly the things for which
induction makes sense.

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construction of R from Q is -different- from the rest:
it's topological, not pure algebra

