Let $\pi: E \to M$ a smooth vector bundle over M. If $(M,g^{M})$ and $(E,g^{E})$ are complete manifolds. Consider $\nabla$ a conection on $E$. We can define these semi-norms
$\displaystyle{\left\|\sigma\right\|_{n}=\sum_{j=0}^{n} \sup_{x\in M}\max_{\{g^{E}_x(v_x^i,v_x^i)=1\}}|\nabla_{v_{x}^{j}}\ldots\nabla_{v_{x}^{1}}\sigma(x)|}$.
How to prove $\displaystyle{d(\sigma,\eta)=\sum_{n=0}^{\infty} \frac{1}{2^n}\frac{\left\| \sigma-\eta \right\|_{n}}{\left\|\sigma-\eta \right\|_{n}+1}}$ is a complete metric in $\Gamma(E)$?