A sub-additive inequality for the volume spectrum

Let $(M,g)$ be a closed Riemannian manifold and $\{{\omega }_p{\}}_{p=1}^{\mathrm{\infty }}$ be the volume spectrum of $(M,g)$. We will show that $\omega_{k+m+1}\leq \omega_k+\omega_m+W$forall$k,m\geq 0$,where$\omega_0=0$and$W$ is the one-parameter Almgren-Pitts width of $(M,g)$. We will also prove the similar inequality for the $\epsilon$-phase-transition spectrum $\{c_{\epsilon }(p){\}}_{p=1}^{\mathrm{\infty }}$ using the Allen-Cahn approach.