Supplement to the article "Approximating multi-dimensional Hamiltonian   flows by billiards": Proof of C$^{0}$ and C$^{r}$ - closeness Theorems
  This supplement to the article "Approximating multi-dimensional Hamiltonian flows by billiards" contains the proofs of C$^{0}$ and C$^{r}$ - closeness theorems stated in the article. 
A. Rapoport, V. Rom-Kedar and D. Turaev