Sequential compactness of smooth functions Suppose I have a sequence $u_n$ of smooth functions on the $N$-dimensional reals. If $\|D^{\alpha}u_n\|_{\infty} \leq C_{\alpha}$ for all multi-indices $\alpha$, then is it possible to deduce that there exists a subsequence with smooth limit? I think there must be if we apply Arzelà-Ascoli infinitely many times, but don't know how to make this rigorous. I suspect contradiction but can't see how.
 A: 
You can find a function $u\in C^\infty\left(\Bbb R^N\right)$ and a subsequence $u_{n_k}$ such that $u_{n_k}$ converges uniformly to $u$ on compact subsets of $\Bbb R^N$ and all the derivatives $D^\alpha u_{n_k}$ converge to $D^\alpha u$ uniformly on compact subsets.

Let $E_n$ the closed ball of radius $n$.
Since $\{u_k\}_{k\in\Bbb N}$ is uniformly bounded in $C(E_1)$ and uniformly lipschitz, by Ascoli-Arzelà theorem there exists a sub-sequence $\{u_{k_h}\}_{h\in\Bbb N}=\{u^{1}_{k}\}_{k\in\Bbb N}$ such that $u^1_k\stackrel{C({E_1})}{\longrightarrow} u\in C(E_1)$.
Since $\{D^\alpha u^1_k\}$ is uniformly bounded and uniformly lipschitz on $E_2$ for all $\lvert\alpha\rvert=1$, there exists a subsequence $\{u^1_{k_h}\}_{h\in\Bbb N}=\{u^2_k\}_{k\in\Bbb N}$ such that $\forall \lvert\alpha\rvert=1,\ D^\alpha(u^2_k)\stackrel{C(E_2)}{\longrightarrow}g_\alpha\in C(B_2)$.
Iterating the procedure, we can find sequences $\{u_k^m\}_{k\in\Bbb N}$ such that:


*

*$u^0_k=u_k$

*$\{u_k^{m+1}\}_{k\in\Bbb N}$ is a subsequence of $\{u_k^m\}_{k\in\Bbb N}$

*For all $\lvert \alpha\rvert=m-1$, $D^\alpha u^m_k\stackrel{k\to\infty}{\longrightarrow} g_\alpha\in C(E_m)$ uniformly on $E_m$.


Now, consider $\{f_k\}_{k\in\Bbb N}=\{u^k_k\}_{k\in\Bbb N}$. We'll show that, for all $n$, $\{f_k|_{E_n}\}$ converges uniformly on $E_n$ to a function $u\in C^{n-1}(E_n^\circ)$.
Consider all the $\alpha$ such that $\lvert\alpha\rvert=n-1\ge 1$. Since:


*

*$D^\alpha u^k_k\to_k g_\alpha$ uniformly on $E_n$

*$D^\beta u^k_k(0)\to_k g_\beta(0)$ for all $\lvert \beta\rvert\le n-2$

*$E_n$ and $E_n^\circ$ are simply connected
It follows that $f_k|_{E_n}$ converges uniformly on $E_n$ to a function $u\in C(E_n)$ such that:


*

*$D^\alpha u(x)=g_\alpha(x)$ for all $x\in E^\circ_n$ and $\lvert\alpha\rvert=n-1$

*$D^\beta u$ extends $g_\beta$ to $E^\circ_n$ for all $\lvert \beta\rvert\le n-2$ and $D^\beta u^k_k\to_k D^\beta u$ uniformly on $E^\circ_n$.
The fact that $f_k\to_k f\in C^\infty\left(\Bbb R^N\right)$ uniformly on compact subsets of $\Bbb R^N$ follows easily form these observations and uniqueness of uniform limits.
