Let $n \in \mathbb N$, $n \ge 2$ and let $S_n$ be the symmetric group on $n$ elements.
I will call for shortness $I_n := \{1 , \ldots , n\} \subset \mathbb {N}$. Fix $i_0 \in I_n$ and consider the following statement:
$\forall \sigma \in S_n, \quad \exists i \in I_n, \quad i \ge i_0 \quad \mathrm{ s.t.} \quad \sigma(i)\le i_0$
I think it is true, it seems like an application of pigeonhole but I don't manage to write a formal and clear proof. I also tried with reductio ad absurdum, unsuccessfully.
What would you suggest? Do you think the statement is true? Thanks.