Suppose that $F ⊂ P(n)$ is a set system containing no chain with $k + 1$ sets.
Prove that $\sum\limits_{r=1}^n \frac{|F_{r}|}{n \choose r} ≤ k$, where $F_{i} = F \cap [n]^{(i)}$ for each i.
($[n]^{(i)}$ is the set of all subsets of $[n]$ which have size $i$)
It's clearly very similar to the LYM equality but I can't seem to work out whether I'm supposed to use the LYM equality or adapt the proof for it.