= Solution
Choose the <continuously differentiable functions>
$$
f_n(x)=\sqrt{x^2+n^{-2}}.
$$
The estimate $\sqrt{a^2+b^2}\leq a+b$ for $a,b\geq0$ gives
$$
0\leq f_n(x)-|x|\leq\frac1n,\qquad
\|f_n-|\cdot|\|_\infty=\frac1n.
$$
Thus $f_n\to|\cdot|$ by <uniform convergence>, and the <triangle inequality> gives $\|f_n-f_m\|_\infty\leq n^{-1}+m^{-1}$. The <sequence> is a <Cauchy sequence> in $X$, but its unique possible <supremum norm> limit is $x\mapsto|x|$, which is not <differentiable> at zero. That limit is outside $X$, so
$$
\boxed{(C^1([-1,1]),\|\cdot\|_\infty)\text{ is incomplete}.}
$$
Back to article page
