Convergence of local stable manifolds This question is a kind of local version of a previous post (MO224171). 
Let $\ f_n \ $ be a sequence of Morse functions on $\mathbb{R}^d$, converging, in the $C^\infty$-topology, to a limit Morse function $\ f$:
$$ f_n \to f  \ .$$
At any critical point $\ p\ $ of the limit function $\ f$, it can be proved that there exists an open neighborhood $\ U\ $ such that, for $n \geq n_0$, 
$$ f_{n|U} \mbox{ has a unique critical point, $p_n$, of the same Morse index than } p. $$
Moreover, the sequence of these critical points $p_n$ converges to $p$.
On the other hand, let $\ W^s(p,f)\ $ stand for the stable manifold associated to $p$: if $\tau_t(x)$ denotes the integral curve of the gradient of $f$ passing through $x$ at $t=0$, then
$$ W^s(p , f) :=  \left[ \ x \in \mathbb{R}^d \ \colon \  \lim_{t \to \infty} \tau_t (x) = p \ \right] . $$
If $V$ is a neighborhood of $p$, we can also consider the local stable manifold:
$$ W^s_V(p , f) :=  W^s(p , f_{|V}) . $$
My question is:
Does there exist a neighborhood $V$ of $p$ such that the local stable manifolds $\ W^s_V(p_n , f_n)\ $ converge (in some adequate sense) to $W^s_V(p , f)$?
For example, such that the Hausdorff distance $d_H (W^s_V(p_n , f_n) , W^s_V(p , f))$ converge to zero?
I've read in Wikipedia that local stable manifolds vary continuously in a neighborhood of $f$, but couldn't find a proof. Probably this is not difficult for the well versed in dynamical systems or differential topology, but I cannot arrange the proof of the Stable Manifold Theorem to work for a family.
 A: Yes, $p$ is a hyperbolic fixed point of the gradient flow, and local (un)stable manifolds of a hyperbolic fixed persist smoothly under small perturbations (which your functions $f_n$ are, relative to $f$ for $n$ large). I don't have a basic reference at hand, but this is a special case of a normally hyperbolic invariant manifold $M$ consisting of a single point, $M = \{p\}$, see for the NHIM result theorem 4.1(f) in Hirsch, Pugh, Shub: "Invariant manifolds", Springer LNM 538 (1977). The $C^r$ (here $r = \infty$) closeness should be viewed in the way already commented by Pietro Majer: you view $W^{u,s}(p;f|_V)$ as graphs, and then the graphs of $W^{u,s}(p;f_n|_V)$ converge to these in $C^r$ norm.
Moreover, this local result can be extended to any compact neighborhood of $p$ that does not contain other fixed points: simply note that the extended (un)stable manifold is obtained under the gradient flow for a fixed, finite time $t$. Smooth dependence of flows on parameters implies that the flows $\Phi_n^t$ converge to $\Phi^t$, hence the (un)stable manifolds converge too. (Near another fixed point $q$ the problem is that the flow time to pass it blows up.)
