Solution
= Solution
For $n\geq1$, the unit sphere is nonempty and <compact> by the <Heine-Borel theorem>. The <function> $k(x)=\|Ax\|^2$ 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 \b[$A^TA$ always has a real <eigenvalue>], indeed a nonnegative one. This gives existence without assuming the spectral theorem first.