NL-NWO 1989 Problem 5

$\forall k\in N \,\, \exists n(k) \in N, a(k):0<a(k)<1 [(1+\sqrt2)^{2k+1}=n(k)+a(k)]$ Prove: $(n(k) + a(k))a(k) = 1$ , for all $k \in N$ , and calculate $\lim_{k \to \infty }a(k)$