product of ideals in decending central series

Question

Let $L$ be a Lie algebra. Let $C^n(L)$ be defined by: $C^0(L) = L$, $C^k(L) = [L,C^{k-1}(L)]$ for $k \geq 1$. Then how can I show that $[C^r(L),C^s(L)] \subseteq C^{r+s}(L)$ for all $r,s \in \mathbf{N}$?

I have tried fixing $s$ and proceeding by induction on $r$ without much luck.

Answer 1

Hint: Rewrite the Jacobi identity as $$[[yz]x] = [y[zx]] + [z[xy]]$$