Cobordism independence of Grassmann manifolds

This note proves that, for $F=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k<n$ and having real dimension $d$, constitute a linearly independent set in the unoriented bordism group ${\mathfrak{N}}_d$ regarded as a ${\mathbb{Z}}_2$-vector space.