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.

same$D^s$? (I looked at a proof of the Stable Manifold Theorem, and tried to arrange precisely this idea, but I couldn't avoid the $D^s_n$ of the $F_n$ from becoming arbitrary small!). $\endgroup$