Let $s_i = (i, i+1)$ be the $n-1$ transpositions that generate $S_n$. The (strong) Bruhat order says that $u \le v$ if for some reduced word $w$ composed of $s_i$ representing $v$, there is a (not necessarily contiguous) subword of $w$ that represents $u$.
My question is, is it true that if $u\le v$ under the definition above, then for EVERY reduced word $w$ representing $v$, we can find a subword representing $u$? I have seen this stated as a definition too but do not know why they are equivalent.
Generalizations to Coxeter groups are welcome, but I prefer an answer that just focuses on $S_n$.