Bounded sequences ℓ∞

Example

Let $\ell^\infty$ be the set of bounded sequences $x = (x_n)_{n \ge 1}$ of real (or complex) numbers, with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$. Then $\ell^\infty$ is a sequence space and a Banach space; this is the case $p = \infty$ of the spaces ℓ^p.

Check of completeness: let $(x^{(j)})$ be a Cauchy sequence in $\ell^\infty$. For each $n$, $\abs{x^{(j)}_n - x^{(i)}_n} \le \norm{x^{(j)} - x^{(i)}}_\infty$, so $(x^{(j)}_n)_j$ is a Cauchy sequence of numbers with a limit $x_n$. Given $\varepsilon > 0$, choose $J$ with $\norm{x^{(j)} - x^{(i)}}_\infty \le \varepsilon$ for $i, j \ge J$. Letting $i \to \infty$ gives $\abs{x^{(j)}_n - x_n} \le \varepsilon$ for all $n$ and all $j \ge J$. So $x$ is bounded and $x^{(j)} \to x$ in norm.