= 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>,
$$
U_1^\dagger AU_1=
\begin{pmatrix}
\lambda&*\\
0&C
\end{pmatrix}.
$$
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>.
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