Let $M$ and $N$ be smooth (finite dimensional) manifolds without boundary.
On the set $C^r(M,N)$ we choose the compact-open $C^r$-topology. This topology is defined as follows (I take the definition of the book "Differential Topology" from Hirsch):
Let $(\varphi,U)$ and $(\psi,V)$ be charts on $M$ and $N$ respectively, $K\subset U$ compact, $f\in C^r(M,N)$ with $f(K)\subset V$, $\varepsilon >0$. For this data, define
$\mathcal{N}^r(f,\varphi,U,\psi,V,K,\varepsilon)$
to be the set of all $g\in C^r(M,N)$ with $g(K)\subset V$ and $\sup_{x\in\varphi(K)}||\partial^\alpha_x(\psi\circ g\circ \varphi^{-1})(x)-\partial^\alpha_x(\psi\circ f\circ \varphi^{-1})(x)||_{euclidean}<\varepsilon$
for all multiindices $\alpha$ with $|\alpha|\le r$.
The compact-open $C^r$ topology on $C^r(M,N)$ is then defined to be the topology generated by all the sets $\mathcal{N}^r(f,\varphi,U,\psi,V,K,\varepsilon)$ where $f,\varphi,U,\psi,V,K,\varepsilon$ vary.
What I want is a "neat" criterion to check convergence of a sequence $g_n$ in $C^r(M,N)$ to some $g\in C^r(M,N)$ if $M$ and $N$ are compact.
To that end, is the following true (I call it a lemma)?
lemma: Pick finitely many charts $(U_i,\varphi_i)$ and $(V_i,\psi_i)$ on $M$ and $N$ respectively, $i=1,\ldots k$, $K_i\subset U_i$ compact with $g(K_i)\subset V_i$ and $\bigcup_{i=1}^k K_i=M$. Then $g_n$ converges to $g$ in $C^r(M,N)$ if for every $\varepsilon >0$ there exists a $N(\varepsilon)\in\mathbb{N}$ such that for all $n\ge N(\varepsilon)$ and all $i=1,\ldots k$ we have
$g_n\in \mathcal{N}^r(g,\varphi_i,U_i,\psi_i,V_i,K_i,\varepsilon)$.