When I read the book of Evans, An introduction to Stochastic Differential equations, I am confused at a step of the proof of Ito's chain rule.
Let $u:\mathbb{R}\times[0,T]\rightarrow\mathbb{R},u=u(x,t)$. Suppose $u, u_t=\frac{\partial u}{\partial t},u_x=\frac{\partial u}{\partial x}$ and $u_{xx}=\frac{\partial^2 u}{\partial x^2}$ are continuous.
How to prove that there exists a sequence of polynomials $u^n$ such that $$ u^{n} \rightarrow u, u_{t}^{n} \rightarrow u_{t}, u_{x}^{n} \rightarrow u_{x}, u_{x x}^{n} \rightarrow u_{x x} $$ uniformly on compact subset of $\mathbb{R}\times [0,T]$. Here $u^n$ has the form $$ u_n(x, t)=\sum_{i=1}^{m} f^{i}(x) g^{i}(t) $$ where $f^i,g^i$ are polynomials.
My try:
By Weierstrass theorem, polynomials are dense in $C(I)$. If we consider the compact subset is $[a,b]\times [0,T]$ and suppose $f^i$ is polynomials basis on $[a,b]$ and $g^i$ is polynomial basis on $[0,T]$. Then we have $$ \sup |u(x,t)-\sum_{i=1}^{m} f^{i}(x) g^{i}(t)|\leq \varepsilon. $$ But how to obtain the uniform convergence of partial derivative of $u^n$?