In Matsumoto's "An Introduction to Morse Theory" the author defines a $(C^2,\varepsilon)$-approximation of a smooth function $g:M\to\mathbb{R}$ defined on a compact smooth manifold in the following way :
Take a finite atlas $\{(U_1,\varphi_1),\ldots,(U_s,\varphi_s)\}$ and a family of compacts $K_1,\ldots,K_s$ such that $K_i\subset U_i$ for every $i=1,\ldots,s$ and $$M=K_1\cup\ldots\cup K_s.$$ We say that $f:M\to\mathbb{R}$ is a $(C^2,\varepsilon)$-approximation of $g$ if the following conditions holds over $K_i$ for every $i=1,\ldots,s$ : $$|f(p)-g(p)|<\varepsilon,\quad \left|\frac{\partial f}{\partial x_j}(p)-\frac{\partial g}{\partial x_j}(p)\right|<\varepsilon,\quad \left|\frac{\partial^2 f}{\partial x_j\partial x_k}(p)-\frac{\partial^2 g}{\partial x_j\partial x_k}(p)\right|<\varepsilon,$$ for all $j,k=1,\ldots,m$.
This definition is dependent of the choice of the finite atlas so I would like to substitute it for a notion of $C^2$- convergence of sequences of functions $f_n:M\to\mathbb{R}$, independent of the choice of the finite atlas. Does anybody know such a definition ?