162 4. PROOF OF THE COMPACTNESS THEOREM
In summary we have constructed a sequence of balls such that the fol-
lowing holds:
PROPOSITION 4.22 (Good cover of a Riemannian manifold). Let (Mn, g)
be a complete Riemannian manifold with sectional curvatures [K[ S Co and
injectivity radius inj (0) ;::=:lo> 0. Then there exists a net of points {xa}~=O,
where N E NU { oo} , and a nondecreasing function A ( r) (which depends only
on n, Co and lo) such that
(1) ra = d (xa, 0) is a nondecreasing function of a,
( 2) the balls { B ( xa, A [ ra])} ~=O are disjoint, and
(3) B (0, r) C Ua::;A(r) B (xa, 2A [ra]).3.3. Application to the sequence of manifolds. We are now ready
to apply our construction of the coverings by balls to the sequence Mk.
For each Mk we can construct nets { xk} as in Proposition 4.22. Let r'k ~
d ( x'f:, Ok). 'We next show that the >. [r'kJ are bounded for each a, so we can
find a subsequence which converges.
PROPOSITION 4.23 (Bounds on the distance of the centers to the origins).
r'f: s 2a>. [OJ and r'J: ;::=: >.[OJ for all a -/= 0.
PROOF. The worst case scenario is if the construction is a string of balls
such that the centers are on a distance-minimizing geodesic. In this case,
the distance r'k s >.[OJ + I:~:i 2>. [r~] + >. [r'J:J s 2a>. [OJ. D
COROLLARY 4.24 (Convergence of the distance of the centers to the
origins). There exists a subsequence {(Mk, gk, Ok)}kEN and positive numbers
{r~J aEN such that, for each a, r'J: -t r~ as k -t oo. Hence there is a
function K (a) such that if k ;::=: K (a), then
~), [r~J SA [r'f:J S 2>. [r~].
PROOF. This is a simple diagonalization argument. DCOROLLARY 4.25. Let K' (r) ~ max {K (a) : a SA (r)}. Then if a S
A (r) and k ;::=: K' (r), then
~>. [r~J S >. [r'kJ s 2>. [r~].
PROOF. If k ;::=: K' (r) and as A (r), then k ;::=: K (a). D
We will denote A [r~J by A a. Now define the following collection of balls:DEFINITION 4.26 (Various size balls). Define13a k ='=. B (xa kl 45elOcC A a) '
jja k = B (xa kl 205e20c0 A a) '