My question is regarding a proof of the fact that $H_{c}^{k+n}(M\times \mathbb{R}^{n})\approx H_{c}^{k}(M)$ whenever $M$ is an oriented $m$-manifold whose compactly supported de Rham cohomology groups $H_{c}^{k}(M)$ are finite dimensional. The proof uses Poincaré duality [$H^{k}(M)\approx(H_{c}^{m-k}(M))^{\ast}$] and is straightforward: $$H_{c}^{k+n}(M\times \mathbb{R}^{n})\approx H^{m+n-k-n}(M\times \mathbb{R}^{n})\approx H^{m-k}(M)\approx H_{c}^{k}(M)$$ However, I don't get why $\dim H_{c}^{k+n}(M\times \mathbb{R}^{n})<\infty$ so that $H_{c}^{k+n}(M\times \mathbb{R}^{n})\approx (H_{c}^{k+n}(M\times \mathbb{R}^{n}))^{\ast}\approx H^{m+n-k-n}(M\times \mathbb{R}^{n})$. I've tried defining an injective linear map $H_{c}^{k+n}(M\times \mathbb{R}^{n})\to H_{c}^{k}(M)$ in order to show that $\dim H_{c}^{k}(M)<\infty $ implies $\dim H_{c}^{k+n}(M\times \mathbb{R}^{n})<\infty$, but I did not succeed.
Any help would be appreciated.