= Solution
Let $L$ be the tridiagonal matrix with diagonal entries $-2$ and adjacent off-diagonal entries $1$ on the interior grid points. The finite-interval scheme is
$$
\left(I-\frac\mu2L\right)u^{n+1}
=\left(I+\frac\mu2L\right)u^n.
$$
For $J$ interior points, the <dirichlet discrete Laplacian> has eigenvalues
$$
\lambda_j=-4\sin^2\left(\frac{j\pi}{2(J+1)}\right)<0,
\qquad 1\leq j\leq J.
$$
The corresponding eigenvalues of the amplification matrix are
$$
q_j=\frac{1+(\mu/2)\lambda_j}
{1-(\mu/2)\lambda_j}.
$$
Since $\lambda_j<0$ and $\mu\geq0$,
$$
|q_j|\leq1
$$
for every $j$. The matrix is real symmetric in the sine eigenbasis, so its induced $2$-norm is the largest $|q_j|$. This proves the <crank-Nicolson stability on a finite Dirichlet interval> and gives
$$
\boxed{\text{stability for every }\mu>0}.
$$
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