= Solution
The characteristic <polynomial> is $\lambda^2+2\gamma\lambda+k^2=0$. Since $0<\gamma<|k|$,
$$
\lambda_\pm=-\gamma\pm i\sqrt{k^2-\gamma^2}.
$$
Forward <Euler method> has amplification eigenvalues $g_\pm=1+h\lambda_\pm$. Their squared modulus is
$$
|g_\pm|^2=(1-\gamma h)^2+(k^2-\gamma^2)h^2=1-2\gamma h+k^2h^2.
$$
Thus
$$
\boxed{0\le h\le\frac{2\gamma}{k^2}.}
$$
Strict inequality gives numerical damping; at the upper boundary the amplitude is constant despite physical decay, so that boundary is a poor accuracy choice. In the undamped limit $\gamma=0$, $|g|^2=1+k^2h^2>1$ for every $h>0$: forward <Euler method> is unstable for any nonzero step on an undamped <harmonic oscillator>.
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