Let $\mathcal{C}^{1}([a,b],\mathbb{R})$ the vectorial space of functions $f:[a,b]\to\mathbb{R}$ of class $\mathcal{C}^{1}$, with the norm $\vert\vert f\vert\vert=\displaystyle\sup_{x\in[a,b]}(\vert f(x)\vert+\vert f'(x)\vert)$. Show that if $\displaystyle\lim_{n\to\infty}{f_{n}}=f$ on $\mathcal{C}^{1}([a,b],\mathbb{R})$ means that $f_{n}\to f$ and $f'_{n}\to f$ uniformly on $[a,b]$.
I know that if $f_{n}$ converges uniformly to $f$, then $\vert\vert f_{n}-f\vert\vert=\displaystyle\sup_{x\in[a,b]}\{\vert f_{n}(x)-f(x)\vert+\vert f'_{n}(x)-f'(x)\vert\}$ $\leq\displaystyle\sup_{x\in[a,b]}\vert f_{n}(x)-f(x)\vert+\displaystyle\sup_{x\in[a,b]}\vert f'_{n}(x)-f'(x)\vert=0$,
Therefore, $\vert f_{n}(x)-f(x)\vert=0$ and $\vert f'_{n}(x)-f'(x)\vert=0$,follow that $f_{n}\to f$ and $f'_{n}\to f'$ uniformly. Is this correct?? Regards!