= Solution
Put $k=\Delta t>0$ and retain the same <matrix> $L$. The <Crank--Nicolson method> is
$$
U^{n+1}-U^n=\frac{k}{2}L(U^{n+1}+U^n).
$$
Take the real mesh-weighted inner product with $U^{n+1}+U^n$. The cross terms cancel, and the identity from part (b) gives
$$
\|U^{n+1}\|_d^2-\|U^n\|_d^2
=\frac{k}{2}\operatorname{Re}\langle L(U^{n+1}+U^n),U^{n+1}+U^n\rangle_d
\leq0.
$$
The implicit system is uniquely solvable: if $(I-kL/2)V=0$, then
$$
\|V\|_d^2=\frac{k}{2}\operatorname{Re}\langle LV,V\rangle_d\leq0,
$$
so $V=0$. Therefore its <dissipative Cayley-transform contraction> satisfies
$$
\boxed{\left\|\left(I-\frac{k}{2}L\right)^{-1}
\left(I+\frac{k}{2}L\right)\right\|_2\leq1.}
$$
Iterating proves \b[unconditional stability for every $\mu=k/d^2>0$]; the perturbation bound is one and does not depend on the mesh or time-step ratio.
The printed hint's exponential estimate is valid with the logarithmic norm, but its proposed bound by $r(k\alpha[L])$ is not a general inheritance principle. For the trapezoidal <stability function> $r(z)=(1+z/2)/(1-z/2)$, that real number can even be negative when $k\alpha[L]<-2$, whereas a norm is nonnegative. The direct energy proof above establishes the required result without that assertion.
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