We have an integer $N$ and $k$ integers $n_1 \geq n_2 \geq \ldots \geq n_k$. We also have a random permutation $\pi$ which permutes the indices of $[1, k]$.
I would like to prove that we have the following inequality for any random permutation $\pi$:
$$\lceil N/n_1 \rceil + \lceil N/(n_1n_2) \rceil + \cdots + \lceil N/(n_1n_2\ldots n_k) \rceil \leq \lceil N/n_{\pi(1)} \rceil + \lceil N/(n_{\pi(1)}n_{\pi(2)}) \rceil + \cdots + \lceil N/(n_{\pi(1)}n_{\pi(2)}\ldots n_{\pi(k)}) \rceil$$
Is it obvious ?
Thank you very much.
Notice that for each permutation $\pi$ we have $n_1 \geq n_{\pi(1)}$, $n_1n_2 \geq n_{\pi(1)}n_{\pi(2)}$, and so on. From this it follows:
\begin{align} \lceil N/n_1 \rceil &\leq \lceil N/n_{\pi(1)}\rceil\\ \lceil N/(n_1n_2) \rceil &\leq \lceil N/(n_{\pi(1)}n_{\pi(2)})\rceil\\ &\vdots \\ \lceil N/(n_1n_2\ldots n_k) \rceil &\leq \lceil N/(n_{\pi(1)}n_{\pi(2)}\ldots n_{\pi(k)})\rceil\\ \end{align}
Summing left sides and right sides gives the desired inequality.