In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume", http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf, the authors make the assertion:

Assertion 0.1. If $M^n$ is complete with sectional curvature $|K|\le 1$, and $Vol(M)<\infty$. Then $M^n$ admits an exhaustion by compact manifolds with smooth boundary, $M^n_k$, such that $Vol(\partial M^n_k)\to 0$ and for which the second fundamental forms $II(\partial M^n_k)$ are uniformly bounded.

As indicated in their paper, the proof depends on a generalization of the argument of Gromov's almost flat manifold paper, which will not be treated in this paper. I am wondering, is there any reference that gives a proof of this assertion?