 Solution

For $n\ge 1$, the unit sphere is nonempty and compact by the Heine-Borel theorem. The function $k(x)=\Vert Ax{\Vert }^2$ is continuous and therefore attains a minimum and a maximum on it by the extreme value theorem. Either point supplies a nonzero eigenvector in part (ii), with a real multiplier. Thus $A^{T}A$ always has a real eigenvalue, indeed a nonnegative one. This gives existence without assuming the spectral theorem first.