The Veronese map is a well-defined closed immersion

Scaling $x$ by $\lambda$ scales every degree-$d$ monomial by ${\lambda }^d$; not all monomials vanish because some $x_i\mathrm{\ne }0$. Thus the homogeneous-coordinate criterion makes ${\nu }_{n,d}$ a morphism.

On the open where $x_i^d\mathrm{\ne }0$, the ratios $x_j{x}_i^{d-1}\mathrm{/}{x}_i^d=x_j\mathrm{/}{x}_i$ recover the usual affine coordinates. These chart inverses show injectivity and regular local inverse maps.

The relations $Z_{\alpha }{Z}_{\beta }-Z_{\gamma }{Z}_{\delta }$ whenever $\alpha +\beta =\gamma +\delta$ vanish on monomial coordinates. Repeatedly exchanging one variable between two degree-$d$ multi-indices gives $Z_{\alpha }^d={\prod }_j{Z}_{d{e}_j}^{{\alpha }_j}$. Thus every point satisfying the relations has some $Z_{d{e}_i}\mathrm{\ne }0$: otherwise every $Z_{\alpha }$ would vanish. On this chart the same exchanges give $Z_{\alpha }\mathrm{/}{Z}_{d{e}_i}={\prod }_j{\left(Z_{(d-1)e_i+e_j}\mathrm{/}{Z}_{d{e}_i}\right)}^{{\alpha }_j}$. The chart is therefore precisely the image of the affine chart $x_i\mathrm{\ne }0$, with inverse coordinates $x_j\mathrm{/}{x}_i=Z_{(d-1)e_i+e_j}\mathrm{/}{Z}_{d{e}_i}$. The common zero locus of the homogeneous relations is exactly the image, so the image is closed and these chart inverses make the map a closed immersion.