Suppose that $f \in C^{\infty}_c ( \mathbb{R}^2 )$, i.e. $f$ is a $C^{\infty}$ function with compact support defined on $\mathbb{R}^2$. The following link

tells us that $f$ can be approximated by a sequence of functions $(f_n)$ in $C^{\infty}_c ( \mathbb{R}) \otimes C^{\infty}_c ( \mathbb{R})$ in the infinity norm, where each $f_n$ is of the form

$$ f_n( x,y) = \sum_{i=1}^{k_n} a^{(n)}_i (x) b^{(n)}_i (y), $$

for some functions $a^{(n)}_i, b^{(n)}_i \in C^{\infty}_c ( \mathbb{R})$, $1 \leq i \leq k_n$.

I wonder whether it is possible to construct such a sequence of functions $(f_n)$ such that $$ \sup_{n \in \mathbb{N}, 1 \leq i \leq k_n} \big\| a^{(n)}_i \big\|_{\infty} < + \infty, \quad \quad \sup_{n \in \mathbb{N}, 1 \leq i \leq k_n} \big\| b^{(n)}_i \big\|_{\infty} < + \infty, $$ and $$ \sup_{n \in \mathbb{N}, 1 \leq i \leq k_n} \big\| \big( a^{(n)}_i \big)' \big\|_{\infty} < + \infty, \quad \quad \sup_{n \in \mathbb{N}, 1 \leq i \leq k_n} \big\| \big( b^{(n)}_i \big)' \big\|_{\infty} < + \infty,$$ where as usual, $h'$ denotes the derivative to the function $h$.

This proposition would be very useful for my proof. But is this statement true? I am unable to prove this by the standard construction of the sequence $(f_n)$ using the Stone-Weierstrass theorem. Any ideas?