= Solution
The original PDF has $u_m^{n+1}$ on the left of the update; the converted TeX omits its $+1$. Use the <Forward Euler method> with $k=\Delta t\geq0$ and $\mu=k/h_x^2$. The <explicit time stepping for bounded reaction diffusion> update <matrix> is
$$
E_h=H_h+kV_h,\qquad H_h=I+kD_h.
$$
When $0\leq\mu\leq1/2$, the heat update $H_h$ has nonnegative stencil weights $\mu,1-2\mu,\mu$ and row sums at most one. Hence its induced maximum <operator norm> is at most one. It is also a <symmetric matrix> with <eigenvalues>
$$
1-4\mu\sin^2\frac{j\pi}{2(M+1)},\qquad 1\leq j\leq M,
$$
all in $[-1,1]$, so its discrete <L2 norm> operator bound is at most one too. Let $A_*=\max(|a_0|,|a_+|)$. In either of these <norms>,
$$
\|E_h\|\leq\|H_h\|+k\|V_h\|\leq1+kA_*.
$$
Iteration gives the mesh-uniform <stability> estimate
$$
\boxed{\|u^n\|\leq(1+kA_*)^n\|u^0\|\leq e^{A_*nk}\|u^0\|,
\qquad nk\leq T.}
$$
The same estimate applies to differences of solutions. Adding a per-step defect $kr^n$ gives $\|e^n\|\leq e^{A_*T}(\|e^0\|+k\sum_{j<n}\|r^j\|)$.
This proof only requires $\mu\leq1/2$ and bounded reaction coefficients. It does not assume the full update $E_h$ is nonnegative: a negative $a_m$ can make its middle stencil weight negative when $\mu=1/2$. Nor does it claim contractivity or an all-time bound for every coefficient. The required <stability> is a bound uniform in the discretization on fixed finite intervals.
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