With what minimum acceleration can a fireman slides down a rope while breaking strength of the rope is $\frac{2}{3}$of his weight

If man slides down with some acceleration then its apparent weight decreases. For critical condition rope can bear only 2/3 of his weight. If a is the minimum acceleration then,
Tension in the rope $=\mathrm{m}(\mathrm{g}-\mathrm{a})$=Breaking strength
$\Rightarrow \mathrm{m}(\mathrm{g}-\mathrm{a})=\frac{2}{3}\mathrm{mg}\Rightarrow \mathrm{a}=\mathrm{g}-\frac{2\mathrm{g}}{3}=\frac{\mathrm{g}}{3}$