Finite-chart Euclidean embedding of a compact smooth manifold ($\Psi (x)=({\rho }_i(x),{\rho }_i(x){\chi }_i(x){)}_i$)

Choose finitely many coordinate charts on a compact manifold and smooth functions supported in them, positive on a cover. Extend the displayed chart products by zero. A positive first coordinate and its matching chart coordinates separate points, while their derivatives detect every nonzero tangent vector. Thus the map is an injective immersion; compactness makes it a smooth embedding. Extra zero coordinates make its codimension positive if required.