I came across a property of monomials in a Schubert polynomial in Lascoux's book:

http://www-igm.univ-mlv.fr/~al/ARTICLES/CoursYGKM.pdf

page 62, footnote 4. The property is as follows.

Let us adopt the convention that Schubert polynomials in $\mathbb{Z}[x_1, \dots, x_n]$ are indexed by $\mathbb{N}^n$. Then for $v=(v_1, \dots, v_n)\in \mathbb{N}^n$, a monomial $x_1^{u_1}\dots x_n^{u_n}$ appears in $Y_v$ only if: $$u_n\leq v_n,~u_n+u_{n-1}\leq v_n+v_{n-1},~\dots,~u_n+u_{n-1}+\dots+u_1\leq v_n+v_{n-1}+\dots +v_1$$

Lascoux mentioned that "it is easy to prove by induction" that this holds. But I've tried and found no obvious clue for such a inductive proof.

So I would be very grateful if anyone gives some hint for this (or point out a reference for this)? Thank you.