Let $\mathcal{F}\subset \mathcal{P}[n]$ be a set system such that each chain in it has at most $k$ sets. Prove that \begin{equation}\sum_{i=0}^n\frac{|\mathcal{F}_i|}{{n\choose i}}\leq k,\end{equation} where $\mathcal{F}_i=\mathcal{F}\cap [n]^{(i)}$.
I am not really sure how to do this. The LYM inequality says if $\mathcal{F}$ is an antichain then the sum on the LHS is bounded by $1$. But for a general set system this seems harder. My thought is that perhaps we can show that at most $1/k$ of the set system elements can be antichains?