Past exam of the functional analysis I course of the University of Oxford / 2023 / 1 / d / ii

Choose the continuously differentiable functionsThe estimate $\sqrt{a^2+b^2}\le a+b$ for $a,b\ge 0$ givesThus $f_n\to |\cdot |$ by uniform convergence, and the triangle inequality gives $\Vert f_n-f_m{\Vert }_{\infty }\le n^{-1}+m^{-1}$. 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