Convergence of a sequence of sections of a bundle is defined as follows:

**Definition:** Let $E$ be a vector bundle over a manifold $M$, and let metrics $g$ and connections $∇$ be given on $E$ and on $TM$. Let $Ω ⊂ M$ be an open set with compact closure $\bar{Ω}$ in $M$, and let $(ξ_k)$ be a
sequence of sections of $E$. For any $p ≥ 0$ we say that $ξ_k$ converges in $C^p$ to
$ξ_∞ ∈ Γ(E\big|_{\bar{\Omega}})$
if for every $ε > 0$ there exists $k_0 = k_0(ε)$ such that
$$\sup_{0\leq |\alpha | \leq p}\sup_{x\in \bar{\Omega}}|\nabla^{\alpha}(\xi_k -\xi_\infty)|_{g}<\varepsilon$$

whenever $k > k_0$. $\nabla^\alpha$ is the covariant derivative corresponding to the multi-index $α$.

**Question:** In the book "The Ricci Flow in Riemannian Geometry" by Ben Andrews and Christopher Hopper, is written:
**Note that since we are working on a compact set, the choice of metric and
connection on $E$ and $TM$ have no affect on the convergence.**

I can't understand why the sentence is true. Can someone help me? Thanks in advance.