= Solution
When $a=0$, the explicit <amplification factor> becomes
$$
G_E(\theta)=1-i\nu\sin\theta,\qquad |G_E(\theta)|^2=1+\nu^2\sin^2\theta.
$$
A single <Fourier mode> has constant modulus in space and is not in the <l2 sequence space>. Instead choose a whole interval of frequencies on which there is definite growth. Let
$$
E=[\pi/4,3\pi/4]\cup[-3\pi/4,-\pi/4],\qquad\widehat U^0(\theta)=\mathbf1_E(\theta).
$$
The <inverse Fourier transform> here gives the real <sequence>
$$
U_j^0=\begin{cases}
\dfrac{\sin(3\pi j/4)-\sin(\pi j/4)}{\pi j},&j\ne0,\\
\dfrac12,&j=0.
\end{cases}
$$
By the <Parseval identity>, its squared <norm> is $\|U^0\|_{\ell_2}^2=\Delta x\,|E|/(2\pi)=\Delta x/2$, finite and strictly positive. These data are also compatible with sampling a bounded <continuous function>: use
$$
u_0(x)=\begin{cases}
\dfrac{\sin(3\pi x/(4\Delta x))-\sin(\pi x/(4\Delta x))}{\pi x/\Delta x},&x\ne0,\\
\dfrac12,&x=0.
\end{cases}
$$
The value at zero is the <limit> of the displayed quotient, and $u_0(j\Delta x)=U_j^0$.
On $E$ we have $\sin^2\theta\geq1/2$. Therefore the transformed solution and the <Parseval identity> give
$$
\begin{aligned}
\|U^m\|_{\ell_2}^2
&=\frac{\Delta x}{2\pi}\int_E(1+\nu^2\sin^2\theta)^m\,d\theta\\
&\geq(1+\nu^2/2)^m\|U^0\|_{\ell_2}^2.
\end{aligned}
$$
For each fixed pair $\Delta x,\Delta t>0$, take
$$
\boxed{c(\Delta x,\Delta t)=\sqrt{1+\frac12\left(\frac{\Delta t}{\Delta x}\right)^2}-1>0.}
$$
Taking square roots proves $\|U^m\|_{\ell_2}\geq(1+c)^m\|U^0\|_{\ell_2}$ for every $m\geq1$. In particular the <norm> already increases at the first step, so \b[no positive mesh pair gives practical stability] in the contraction sense of (b).
The native PDF places the existence of $c>0$ before the universal quantifier over mesh pairs. If this means a single positive $c$ independent of $\Delta x,\Delta t$, that assertion is false for nonzero data. Indeed, for every datum the same <Fourier symbol> gives
$$
\|U^1\|_{\ell_2}\leq\sqrt{1+\nu^2}\,\|U^0\|_{\ell_2}.
$$
For any proposed fixed $c>0$, choose $0<\nu<\sqrt{(1+c)^2-1}$. The upper bound then contradicts the requested lower bound at $m=1$. The correct quantifiers are $\forall\Delta x,\Delta t>0\ \exists c=c(\Delta x,\Delta t)>0$, with nonzero square-summable data as constructed above.
Finally, failure of this stronger <practical stability of a finite difference scheme> should not be confused with every possible refinement notion of <numerical stability>. If $\Delta t/(\Delta x)^2\leq K$, then $m\Delta t\leq T$ gives
$$
|G_E(\theta)|^m\leq\exp(m\nu^2/2)\leq\exp(KT/2).
$$
Thus a finite-time mesh-uniform bound can hold under this severe refinement even though contraction fails on every positive mesh. This is the distinction recorded in <centered-advection refinement-dependent stability>.
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