= Solution
Using the <semidiscrete Fourier transform> from (b), the explicit scheme has <amplification factor>
$$
G_E(\theta)=1-4a\mu\sin^2(\theta/2)-i\nu\sin\theta.
$$
Set $r=a\mu$ and $s=\sin^2(\theta/2)\in[0,1]$, so $\sin^2\theta=4s(1-s)$. Direct expansion gives
$$
|G_E|^2=(1-4rs)^2+4\nu^2s(1-s),\qquad
|G_E|^2-1=4s\left[(\nu^2-2r)+(4r^2-\nu^2)s\right].
$$
The expression in square brackets is an <affine function> of $s$. At the two endpoints its values are
$$
\nu^2-2r\leq0,\qquad 4r^2-2r=2r(2r-1)\leq0,
$$
using precisely $\nu^2\leq2r\leq1$. Every intermediate value is a <convex combination> of these endpoints, so $|G_E|^2\leq1$ for every frequency. The <Parseval identity> proves
$$
\boxed{\|U^m\|_{\ell_2}\leq\|U^0\|_{\ell_2}\quad(m\geq0),\qquad\nu^2\leq2a\mu\leq1.}
$$
Thus the scheme is <practically stable> under the given condition. For $a>0$ it is equivalent to $\Delta t\leq2a$ and $\Delta t\leq(\Delta x)^2/(2a)$. This is the <Forward Euler stability for centered advection-diffusion> calculation with the diffusion coefficient retained. The proof does not require all three stencil weights to be nonnegative.
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