= Solution
The <Gershgorin circle theorem> says that every <eigenvalue> of a square <matrix> lies in at least one disc centred at a diagonal entry with radius the sum of the absolute values of the off-diagonal entries in that row.
For interior unknowns write the update as $u^{n+1}=Bu^n$. The <matrix> is real symmetric and tridiagonal, with diagonal $1-\mu(a_{m-1/2}+a_{m+1/2})$ and adjacent entries $\mu a_{m+1/2}$. Each Gershgorin radius is at most $\mu(a_{m-1/2}+a_{m+1/2})$. Its real interval is therefore contained in
$$
[1-2\mu(a_{m-1/2}+a_{m+1/2}),1]\subset[1-4\mu a_+,1].
$$
For \b[$0<\mu\leq1/(2a_+)$], every <eigenvalue> lies in $[-1,1]$. Symmetry gives an orthogonal diagonalization, so $\|B^n\|_2\leq1$ for every $n$: errors in initial data cannot grow. This is stability in the discrete $\ell^2$ norm, also in its $h$-weighted version. One can additionally see maximum-norm stability directly: the update weights are nonnegative with row sums at most one once the zero boundary unknowns are removed. Hence $\|B\|_\infty\leq1$.
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