Past exam of the analysis course of the University of Oxford / 2011 / 1 / b / i / Solution

Show that $c$ is a closed linear subspace of ${\ell }^{\infty }$. Suppose $a^{(j)}\in c$ and $\Vert a^{(j)}-a{\Vert }_{\infty }\to 0$. Given $\epsilon >0$, choose $j$ with uniform error below $\epsilon /3$. Since $a^{(j)}$ converges, choose $N$ such that $|a_m^{(j)}-a_n^{(j)}|<\epsilon /3$ for $m,n\ge N$. ThenThus $(a_n)$ is a real Cauchy sequence and converges, so $a\in c$. A closed linear subspace of a Banach space is complete in the inherited norm. Therefore $\boxed{(c,\Vert \cdot {\Vert }_{\infty })\text{ is Banach}}$.