§ 4. Differential Manifolds
257
However, (13) and (15) show that
D n+1 (t, z) ≤
U n (u, z)Y n (u, z) − U (u, z)Y (u, z) du +
+
V n (u, z) − V (u, z) du.
≤
U n (u, z) − U (u, z) . Y (u, z) du +
+
U n (u, z) .D n (u, z)du +
+
V n (u, z) − V (u, z) du
with extended unoriented integrals over the interval I(t) with endpoints 0
and t; as I(s) ⊂ I(t) for s ∈ I(t), the right hand side is even an upper bound
for D n+1 (s, z) for all s ∈ I(t). If k is a constant upper bound for Y (u, z) and
the U n (u, z) – they converge uniformly – for |u| ≤ a
and |z| ≤ c, then
D n+1 (s, z) ≤ k
U n (u, z) − U (u, z) du + k
D n (u, z)du +
+
V n (u, z) − V (u, z) du ,
where integration is over I(t), and not only over I(s).
Let r > 0. There exists N = N (r) such that
U n (u, z) − U (u, z) ≤ r & V n (u, z) − V (u, z) ≤ r
for n ≥ N , |u| ≤ a
, |z| ≤ c. The previous relation then shows that, for
n ≥ N ,
D n+1 (s, z) ≤ (k + 1)|t| + k
I(t)
D n (u, z)du
(15.17)
for s ∈ I(t). Set A = k + 1 and
Δ n (t) = sup
u ∈ I(t)
|z| ≤ c
D n (u, z) .
The function being integrated on the right hand side of (17) is ≤ Δ n (u).
Taking the sup of the left hand side for s ∈ I(t) and |z| ≤ c, it can be
deduced that
Δ n+1 (t) ≤ rA|t| + A
I(t)
Δ n (u)du
(15.18)
for n ≥ N , and so, iterating,