Consider $C^1[0,1]$ with the metric $d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g')$. Then $C^1$ is complete.
My attempt:
Let $(f_n)_n$ be cauchy in $C^1[0,1]$ wrt $d$. Then $f_n$ and $f'_n$ are cauchy wrt $d_{\infty}$. Since $C[0,1]$ is complete wrt $d_{\infty}$, $f_n$ converges to $f$ pointwise and $f'_n\rightarrow g$ uniformly. Then $f$ is differentiable with $f'(x)=g(x)$ for all $x\in [a,b]$. Let $\epsilon>0$. Choose $N^1$ such that $n\geq N^1$ implies $d_{\infty}(f_n,f)<\frac{\epsilon}{2}$. Similarly, choose $N^2$ such that $n\geq N^2$ implies $d_\infty(f'_n,f')<\epsilon$. Then, for n$\geq max(N^1,N^2)$, $d(f_n,f)= d_{\infty}(f_n,f)+d_{\infty}(f'_n,f')<\epsilon$
Is my attempt correct?