Weight multiplicity decreases away from a dominant weight

If $\mu$ and $\lambda$ are weights of a finite-dimensional representation, $\mu$ is dominant, and $\mu ⪯\lambda$, thenSuccessively subtract a simple root ${\alpha }_i$ for which $⟨\lambda ,{\alpha }_i^{\vee }⟩>0$ and use injectivity of the corresponding ${\mathfrak{sl}}_2$ lowering operator.