I'm reading a research article lately, and got confused about a question. So, the fundamental theorem of Kruskal and Katona states that if each set in a given set system $\mathcal{A}$ has $k$ elements and $|\mathcal{A}|=m$, then the lower shadow of $\mathcal{A}$ is at least as large as the lower shadow of the initial segment of length $m$ of $N^{(k)}$ in the colex order. Here, $N^{(k)}$ is the family of all $k$-element sets of all the natural numbers. Then, the article mentioned that, as an easy consequence of this Kruskal-Katona theorem, we have the following lemma:

If $\mathcal{F}$ is a down-set, $||\mathcal{F}|| \leq || \mathcal{I}(|\mathcal{F}|) ||$. Here, $\mathcal{I}(m)$ is the initial segment of length $m$ of $N^{(<\inf)}$ in the colex order. Here, $N^{(<\inf)}$ is the family of all the finite subsets of all natural numbers. I did not see the reasoning behind this. Could anyone give me a hint of this? Many thanks for your time and attention.

4more comments