= Solution
Use the metric on $M$ induced by its given Euclidean embedding. For two smooth maps define the <C1 metric on a smooth mapping space> by
$$
d_1(f,g)=\sup_{x\in M}|f(x)-g(x)|+
\sup_{x\in M}\|Df_x-Dg_x\|_{\mathrm{op}},
$$
where the <differential> has domain $T_xM$ with the induced <inner product>. Compactness makes both suprema finite. The norm properties give symmetry and the <triangle inequality>, and the first term makes distance zero imply equality of maps. Thus convergence in $d_1$ is <uniform convergence> of both maps and <differentials>.
Because $i$ is an <immersion>, $Di_x$ is injective at every point. The continuous function $|Di_xv|$ on the compact <unit tangent bundle> has a positive minimum $a$. Therefore
$$
|Di_xv|\ge a|v|\qquad(x\in M,\ v\in T_xM).
$$
For all sufficiently large $n$, $\sup_x\|Df_{n,x}-Di_x\|<a/2$, and then
$$
|Df_{n,x}v|\ge |Di_xv|-|(Df_{n,x}-Di_x)v|\ge\frac a2|v|.
$$
Consequently \b[every sufficiently late $f_n$ is an <immersion>]. If $M$ has dimension zero it is a finite set, the <immersion> assertion is automatic, and eventual injectivity follows directly from separation of its finitely many image points; we henceforth treat positive dimension.
For injectivity, choose finitely many coordinate maps $\phi_b$ on convex balls, with smaller coordinate neighborhoods covering $M$ and closures lying inside the larger balls. Write $I_b=i\circ\phi_b$ and $F_{n,b}=f_n\circ\phi_b$. On the finitely many relevant compact coordinate closures, the <differentials> of the $I_b$ have a common positive lower bound $s$, and convergence in $d_1$ gives
$$
|DF_{n,b}(u)v|\ge\frac s2|v|
$$
for all late $n$. The assumed uniform second-derivative bound gives a common bound $L$ for $D^2F_{n,b}$ on those larger coordinate neighborhoods. If the bound is expressed using an intrinsic <Hessian>, the coordinate chain rule also involves first derivatives and derivatives of the fixed charts; their bounds are uniform by C1 convergence and compactness. Thus $L$ can be larger than the printed constant $C$, but is independent of $n$.
For $u,v$ whose segment lies in one such coordinate ball, <Taylor formula with integral remainder> gives
$$
F_{n,b}(v)-F_{n,b}(u)=DF_{n,b}(u)(v-u)+R,
\qquad |R|\le\frac L2|v-u|^2.
$$
Choose $\delta>0$ with $L\delta\le s/2$. Then, whenever $0<|u-v|<\delta$,
$$
|F_{n,b}(v)-F_{n,b}(u)|\ge\frac s2|u-v|-\frac L2|u-v|^2
\ge\frac s4|u-v|>0.
$$
The <Lebesgue number lemma> applied to the finite smaller-chart cover, together with <uniform continuity> of its coordinate inverses on the relevant compact sets, gives $\rho>0$ such that every pair $x,y\in M$ with $0<|x-y|<\rho$ lies in one such chart with coordinate distance below $\delta$. The displayed estimate therefore excludes all sufficiently close pairs of equal images, uniformly in late $n$.
The remaining pairs form the compact set
$$
K_\rho=\{(x,y)\in M\times M:|x-y|\ge\rho\}.
$$
If it is nonempty, injectivity of the <smooth embedding> $i$ gives a positive minimum $b$ of $|i(x)-i(y)|$ on it. <Uniform convergence> makes $\sup_x|f_n(x)-i(x)|<b/3$ for all sufficiently large $n$, and hence on $K_\rho$,
$$
|f_n(x)-f_n(y)|\ge |i(x)-i(y)|-2\sup_z|f_n(z)-i(z)|>b/3.
$$
If $K_\rho$ is empty there is nothing left to check. Combining the near-pair and distant-pair bounds makes $f_n$ injective for every sufficiently late $n$. It is already an <immersion>, and a continuous injection from compact $M$ into a <Hausdorff space> is a <homeomorphism> onto its image. By <compact injective immersion is an embedding>,
$$
\boxed{f_n:M\hookrightarrow\mathbb R^N\text{ is a smooth embedding for all sufficiently large }n.}
$$
This proves the requested <stability of compact smooth embeddings> using the stated second-derivative assumption. That assumption supplies a convenient Taylor constant; the stronger C1 openness result can replace it by the <uniform continuity> of the limiting <differential> on finitely many small coordinate balls.
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