I am reading a paper that claims the following without proof:
Let $\mathbf{1}$ denote the identity function on $\mathbb{R}^d$. Then there exists a sequence of $C^{\infty}_c$ functions $\varphi_n =(\varphi^{(1)}_n , \varphi^{(2)}_n , \ldots, \varphi^{(d)}_n ) : \mathbb{R}^d \to \mathbb{R}^d$ such that on compact sets, $$ \left\{ \begin{array}{ll} \varphi_n \to \mathbf{1}, \\ \frac{\partial}{\partial x_j} \varphi^{(i)}_n \to \delta_{i,j}, \quad \quad i,j \in \{1, \ldots, d\},\\ \frac{\partial^2 }{\partial x_i \partial x_j} \varphi^{(k)} _n \to 0, \quad \quad i,j,k \in \{1, \ldots, d\}, \end{array} \right. $$ uniformly.
Strangely, it further claims that the choice of $\varphi_n$ can be made such that the following conditions are also satisfied:
Without loss of generality, we can assume that there exists a constant $C$ such that for any $n \in \mathbb{N}$ and $v \in \mathbb{R}^d$, $$ \left\{ \begin{array}{ll} \| \varphi_n (v) \| \leq C \| v \|, \\ \big| \frac{\partial}{\partial x_j} \varphi^{(i)}_n (v) \big| \leq C, \quad \quad i,j \in \{1, \ldots, d\},\\ \big| \frac{\partial^2 }{\partial x_i \partial x_j} \varphi^{(k)} _n \big| \leq C, \quad \quad i,j,k \in \{1, \ldots, d\}, \end{array} \right. $$ and that $\varphi_n(v) =v$, for any $n \in \mathbb{N}$ and any $v \in \mathbb{R}^d$ with $\|v \| \leq n$.
The only relevant result that I can recall is the fact that the set $C^{\infty}_c$ is dense in $L^p$. However, this doesn't give any information about the derivatives of $\varphi_n$, nor boundedness of the derivatives. Does this involve any theorem in Sobolev spaces? I have absolutely no idea at the moment.