If I have the sequences of functions $\{f\_j\}\in C^{\infty}(\mathbb{R})$ and $\{g\_j\}\in C^{\infty}(\mathbb{R})$. Assuming also that $f,g\in C^0(\mathbb{R})$.

Under the assumption that $f\_j\rightarrow f$ and $g\_j\rightarrow g$ on every bounded interval $[a,b]$ as $j\rightarrow \infty$ and $f'\_j(x)=g\_j(x)$, then I want to show that $f'(x)=g(x)$.

Thanks in advance