Space of convergent sequences c

Example

Let $c$ be the set of real convergent sequences $x = (x_n)_{n \ge 1}$, with the norm $\norm{x}_\infty = \sup_n \abs{x_n}$. Then $c$ is a Banach space.

Check. Every convergent sequence is bounded, so $c$ is a linear subspace of the Banach space $\ell^\infty$ of bounded sequences. It suffices to show that $c$ is closed in $\ell^\infty$. Let $x^{(k)} \in c$ tend to $x$ in $\ell^\infty$. For all $k$, $m$ and $n$, $$\abs{x_m - x_n} \le 2 \norm{x - x^{(k)}}_\infty + \abs{x^{(k)}_m - x^{(k)}_n}.$$ Choose $k$ to make the first term small, then $m, n$ large to make the second term small. So $x$ is a Cauchy sequence of real numbers, and it converges.