= Solution
Let $T$ be the $M\times M$ <tridiagonal matrix> with diagonal two and adjacent entries minus one, incorporating zero <Dirichlet boundary conditions>. The actual PDF scheme is
$$
U^{n+1}=G_\mu U^n,\qquad G_\mu=(I+\mu T)^{-1}.
$$
An <orthonormal basis> of <eigenvectors> of $T$ has components $\sqrt{2/(M+1)}\sin(jm\pi/(M+1))$, and substitution gives
$$
\lambda_j=2-2\cos\frac{j\pi}{M+1}
=4\sin^2\frac{j\pi}{2(M+1)}>0,\qquad 1\leq j\leq M.
$$
They form an <orthonormal basis> by the <spectral theorem for real symmetric matrices>. Hence $I+\mu T$ is invertible for every $\mu>0$, and the amplification <eigenvalues> are $g_j=(1+\mu\lambda_j)^{-1}\in(0,1)$. For the <discrete L2 norm> $\|U\|_h^2=h\sum_m|U_m|^2$,
$$
\boxed{\|G_\mu^n U\|_h\leq\|U\|_h\quad\text{for all }n\geq0,\ \mu>0.}
$$
This bound has constant one independent of time step and mesh, which is the required <stability of a numerical method>. Therefore \b[every positive <Courant number> is stable]. If a forcing or local-defect sequence is added, the discrete <variation-of-constants formula> gives a bound by the initial <norm> plus the sum of those perturbation <norms>; this also justifies the <global error> conclusion in part (a).
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