Convergence in $C(X,Y)$ with graph topology I can't figure out this point-set topology assertion. Suppose $X$ is paracompact (Hausdorff?) and $Y$ is metrizable with metric $d$. Then a basis of open sets for the graph topology $WO$ on $C(X,Y)$ (take this as the definition if you like) is given by open sets of the form $$U(f,\epsilon) := \{g \in C(X,Y) : d(f(x),g(x)) < \epsilon(x), \epsilon \in C(X,(0,\infty)).$$ 
The assertion is this: 

$f_n \rightarrow f$ in $WO$ iff there is a compact set $K$ such that
  $f_n|_K \rightarrow f|_K$ uniformly and $f_n \equiv f$ off $K$ for all
  but finitely many $n$.

Certainly the condition implies WO convergence, but I can't figure out how to get a converse. Convergence in WO implies that we get uniform convergence on for any subset, but I can't make sense of how to get the other part. The source asserts that we can choose a sequence of points $x_{n_k}$ with no cluster such that $f(x_{n_k}) \neq f_{n_k}(x_{n_k})$. From here, the result is clear enough since we can use a partition of unity to build an $\epsilon$ that makes a problem.
But how do we pick that sequence? If $X$ is hemicompact then I can see how I might proceed, but otherwise I just don't know enough about the compact subsets of $X$. Can anyone who is better at point-set topology than me help out? The source of this statement is Michor's Manifolds of Differentiable Mappings, Lemma 3.4.
 A: So it seems like this is false without an assumption on local compactness. Let $H$ be an infinite dimensional Hilbert space. Then $H$ is paracompact, and if we select a sequence of orthonormal vectors $e_n$, then we can construct functions $f_n$ which are positive on a ball of radius $\frac{1}{n}$ about $\frac{1}{n}e_n$ and zero elsewhere. Then this sequence of functions obviously can't satisfy the theorem, since a compact $K$ cannot have a subspace homeomorphic to an open ball in this Hilbert space. On the other hand, if $\epsilon$ is a positive function on $H$, then set $a = \epsilon(0)$ and $b$ to be the radious of some ball such that $\epsilon > \frac{a}{2}$ on a ball of that radius about $0$. If $\frac{1}{2n} < a,b$, then $f_n$ will be pointwise less than $\epsilon$, and so $f_n \rightarrow 0$ in the WO topology.
On the other hand, a local compactness assumption makes everything work out ok. Since I think the domains of interest are manifolds with corners in Michor, this should probably just be an added hypothesis.
A: @user-j
It is easy to check that Zach L.’s counterexample is OK.
Concerning your reference to [GG] (see the fragments from the pages 42,  43, 44), there is written

the sequence of mappings $f_n$ converge to $f$ (in the Whitney $C^k$ topology) iff there is a compact subset $K$ of $X$ such that $j^kf_n$ converges uniformly to $j^kf$ on $K$ and all but a finite number of the $f_n$'s equal $f$ off $K$. The "if" part is clear, and we shall prove the "only if" part by contradiction. Assume $f_n$ converges to $f$ and that there does not exist a compact set $K$ with the above property. Let $K_1, K_2,\dots$ be a sequence of compact subsets of $X$ such that $K_i\subset \operatorname{Int} (K_{i + 1})$ and $X =\bigcup_{i=1}^\infty K_i$.

For topological spaces the latter assumption is even stronger than hemicompactness. Indeed, according to exercise 3.4.E from [Eng] (referenced to [Are]):

A Hausdorff space $X$ is hemicompact if in the family of all compact subspaces of $X$ ordered by $\subset$ there exists a countable cofinal subfamily.
(a) Prove that every first-countable hemicompact space is locally compact.
(b) Give an example of a countable hemicompact space which is not a $k$-space.
(c) Show that in the realm of second-countable spaces hemicompactness is equivalent to local compactness.
(d) Prove that if the space $\Bbb R^X$ with the compact-open topology is first-countable and $X$ is a Tychonoff space, then $X$ is hemicompact.

References
[Are] Arens R., A topology for spaces of transformations, Ann. Math. 47 (1946) 480-495.
[Eng]  Ryszard Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.
[GG] Martin Golubitsky, Victor Guillemin, Stable Mappings and Their Singularities, Springer-Verlag, 1973.
