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.



On the definition of convergence of a sequence of sections of a bundle
This is just to supply some details to what Rafe Mazzeo wrote.
Let $g\_{i}$ be metrics and $^{\left( i\right) }\nabla$ be connections on $E$
and on $M$ for $i=1,2$. Since $\bar{\Omega}$ is compact,
the uniform equivalence of norms reduces to a local coordinate chart
$(U,\{x^{i}\})$ over which the bundle $E$ is trivialized. In the following,
the constant $C$ may change from line to line. Since $C^{-1}g\_{1}\leq
g\_{2}\leq Cg\_{1}$ (uniform equivalence) on $E$ (fiberwise) and on $M$ for some
$C$, for any $\xi\in\Gamma(E\otimes\bigotimes^{k}T^{\ast}M)$ we have
$|\xi|\_{g\_{1}}\leq C|\xi|\_{g\_{2}}$ (same for $1$ and $2$ switched). Let
$\alpha=(\alpha\_{1},\ldots,\alpha\_{n})$, so that $\nabla^{\alpha}=\nabla
\_{1}^{\alpha\_{1}}\cdots\nabla\_{n}^{\alpha\_{n}}$ (up to uniform equivalence of
norms, we may order it this way since commutators yield curvature and its
derivative terms, which are bounded). Let $\lesssim$ denote $\leq C\cdot$. Now
for $\xi\in\Gamma(E)$,
$$
|^{\left( 1\right) }\nabla^{\alpha}\xi|\_{g\_{1}}\lesssim|^{\left( 1\right)
}\nabla^{\alpha}\xi|\_{g\_{2}}\lesssim|^{\left( 2\right) }\nabla^{\alpha}
\xi|\_{g\_{2}}+|\sum\_{k=0}^{\left\vert \alpha\right\vert -1}{}^{\left(
1\right) }\nabla^{\ast k}\circ(^{\left( 1\right) }\nabla-{}^{\left(
2\right) }\nabla)\circ^{\left( 2\right) }\nabla\_{g\_{2}}^{\ast(\left\vert
\alpha\right\vert -k-1)}|,
$$
where the sum is comprised of linear combinations of $\ell$-th order covariant
derivatives $^{\left( i\right) }\nabla^{\ast\ell}$. Since the sum has only
covariant derivatives of lower order together with (bounded) derivative of the
difference of connections terms, by induction on $p$ we obtain $\sum
\_{\left\vert \alpha\right\vert \leq p}|^{\left( 1\right) }\nabla^{\alpha}
\xi|\_{g\_{1}}\lesssim\sum\_{\left\vert \alpha\right\vert \leq p}|^{\left(
2\right) }\nabla^{\alpha}\xi|\_{g\_{2}}$ over $U$, independent of $\xi$.


