Consider the space $\ell^{\infty}$ of bounded real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ with the norm $\|x\|_{\infty}=\sup _{i}\left|x_{i}\right|$. Show that for every bounded sequence $x^{(n)}$ in $\ell^{\infty}$ there is a subsequence $x^{\left(n_{j}\right)}$ which converges in every coordinate, i.e. the sequence $\left(x_{i}^{\left(n_{j}\right)}\right)_{j=1}^{\infty}$ of real numbers converges for each $i$. Does every bounded sequence in $\ell^{\infty}$ have a convergent subsequence? Justify your answer.

Let $\ell^{1} \subset \ell^{\infty}$ be the subspace of real sequences $x=\left(x_{i}\right)_{i=1}^{\infty}$ such that $\sum_{i=1}^{\infty}\left|x_{i}\right|$ converges. Is $\ell^{1}$ complete in the norm $\|\cdot\|_{\infty}$ (restricted from $\ell^{\infty}$ to $\left.\ell^{1}\right)$ ? Justify your answer.

Suppose that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{\infty}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{1} .$

Suppose now that $\left(x_{i}\right)$ is a real sequence such that, for every $\left(y_{i}\right) \in \ell^{1}$, the series $\sum_{i=1}^{\infty} x_{i} y_{i}$ converges. Show that $\left(x_{i}\right) \in \ell^{\infty} .$