Space of Almost-Zero Sequences is not Closed in 2-Sequence Space

Theorem
Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the normed 2-sequence vector space.

Let $\struct {c_{00}, \norm {\, \cdot \,}_2}$ be the normed vector space of almost-zero sequences.

Then $\struct {c_{00}, \norm {\, \cdot \,}_2}$ is not closed in $\struct {\ell^2, \norm {\, \cdot \,}_2}$.

Proof
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence in $c_{00}$ such that:


 * $\ds x_n := \tuple {1, \frac 1 2, \ldots \frac 1 n, 0, \ldots}$

Let $\ds x := \tuple {1, \frac 1 2, \ldots, \frac 1 n, \ldots}$ with $n \in \N_{>0}$.

We have that $x \in \ell^2 \setminus c_{00}$ where $\setminus$ denotes set difference.

Then:

Pass the limit $n \to \infty$

Then:


 * $\ds \lim_{n \mathop \to \infty} \norm {x_n - x} = 0$

Hence, $\struct {c_{00}, \norm {\, \cdot \,}}$ does not contain its limit points.

By definition, it is not closed.