Convergence of sequences in the strong Whitney topology Let $\{f_n\}_{n \in \mathbb{N}}$ a sequence of $C^r(M, N)$ functions of paracompact manifolds converging to $g$ in the strong Whitney topology. Prove that there exist a compact $K$ of $M$ $f_n$ and $g$ agree every where except on $K$, $\forall n \in \mathbb{N}$
I do not see how to proceed as the strong Whitney topology (as defined in Hirsh' Differential Topology) only guarantees that the $f_n$ (and their derivatives) are $\epsilon_i$-near of $g$ on compacts $K_i$.
Thank you in advance.
 A: To prove this we will go by contradiction, and our main goal now is to construct an open set of $C_S^r(M,N)$ that will not contain an infinite number of $f_n's$,and so contradicting the fact that $f_n\rightarrow g$ in $C_S^r(M,N)$.For this we need to take a compact exhaustion of $M$, that is compact sets $\{K_i\}_{i=1}^{\infty}$ such that their interiors cover $M$ and $K_1\subset int(K_2)\subset K_2 \subset ...$ \.
First we show that there exists a sequence of $x_k's$ in $M$ , and increasing sequences $n_k$ and $l_k$, such that $x_k\in K_{l_k}-int(K_{l_{k-1}})$ and $f_{n_k}(x_k)\neq g(x_k)$. This is done by induction . We have that $K_0= \emptyset$ and take $N=1$, and we have that there exists $x_1\in M$, more precisely in some $K_i$ which we now denote $K_{l_1}$, such that $f_{n_1}(x_1)\neq g(x_1)$, where $n_1>N$.Now suppose we have found $x_1,...x_{k-1}$, $n_{k_1},...,n_{k-1}$ and $l_{k_1},...,l_{k-1}$.Now set $N=n_{k-1}+1$ and consider $K=K_{l_{k-1}}+1$. We have that there exists $x_k\in M-K$ and $n_{k}>N$ such that $f_{n_k}(x_{k})\neq g(x_k)$. We have that $x_k$ belongs to some $K_i$ and so we consider $l_k=i$ and we have that $x_k\in K_{l_k}-int(K_{l_{k-1}})$.\
Now that we have the previous property we now construct the desired open set. First notice that each $x_k\in int(K_{l_k+1})-K_{l_k-2}$, and that these will be open sets.Now consider a coordinate chart around each $x_k$, $(U_k,\phi_k)$, and by doing intersections if needed, we can assume that we have compact sets $K'_k$ and locally finite charts $(U_k,\phi_k)$ such that $K'_k\subset U_k\subset int(K_{l_k+1})-K_{l_k-2}$, furthermore we can assume that there exists charts $(V_k,\psi_k)$ for $g(x_k)$ such that $g(U_k)\subset V_k$. Now since we have that $f_{n_k}(x_k)\neq g(x_k)$ we have that there exists $\epsilon_k$ such that $||\psi_k\circ f_{n_k}(x_k)-\psi_k\circ g(x_k)||>\epsilon_k$. Now putting all of this together consider $\Phi=\{U_k,\psi_k\}_{k=1}^{\infty}, \Psi=\{V_k,\phi_k\}_{k=1}^{\infty}, K=\{K_k\}_{k=1}^{\infty}$ and $\epsilon=\{\epsilon_k\}_{k=1}^{\infty}$, we will have that $N^r(g;\Phi,\Psi,K,\epsilon)$ will be our desired neighborhood.
