Specific reduced expression of elements in $S_n$ ($W_{B_n}$)

Question

I observed that if we consider elements in $S_n$ as permutations, for example we can write $$\sigma=\begin{pmatrix} 1 &2 &3 &4 \\ 4 &3 &2 &1 \\ \end{pmatrix}$$ Then $\sigma$ has a reduced expression $s_3s_2s_1s_3s_2s_3$, which starts with a string $s_3s_2s_1$ that permutes 1 to 4. I have a strong confidnec that this is true for any element in $S_n$ or $W_{B_n}$, however I don't know how to prove that. Any help would be appreciated. To state the property formally: For any $\sigma\in S_n$, it must have a reduced expression starting with $s_{\sigma(1)}s_{\sigma(1)-1}\cdots s_1$