The proof is identical in the two cases. Set $k=k_{i}$ in case (i) and $k=2m_{j}-4$ in case (ii). In case (ii) we work with $\mathbb{Z}_{2}$–invariant forms on the double cover $H^{2m_{j}-4}$.

By scaling we can assume that $\epsilon=1$. It is enough to prove that every closed $2$–form $\eta$ with $\eta=O(\rho^{-3})$ can be written as $\eta=da$ with $|\nabla^{k}a|=O(\rho^{-2-k})$.

Since the restriction of $H^{k}$ to an exterior domain in $\mathbb{R}^{3}$ is diffeomorphic to $(R,\infty)\times\Sigma$ with $\Sigma$ an homology sphere, we can write $\eta=d\rho\wedge\alpha+\beta$ for some $\rho$–dependent $1$–form $\alpha$ and $2$–form $\beta$ on $\Sigma$ with $|\alpha|+|\beta|=O(\rho^{-3})$.

The condition $d\eta=0$ implies $\partial_{\rho}\beta-d_{\Sigma}\alpha=0$. We then define $a=-\int_{\rho}^{\infty}{\alpha}$. The Lemma follows. ∎