 Solution

The proof is by induction on $n$. Choose a unit eigenvector $v$ of $A$, which exists by part (a), and extend it to an orthonormal basis. In that basis,By induction, a unitary change of basis on the last $n-1$ coordinates makes $C$ upper triangular. Extending that change by the identity on the first coordinate leaves the displayed zero block intact. Their product is unitary and makes all of $A$ upper triangular. This proves Schur triangularization.