Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space

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Theorem

Let $\struct {\ell^2, \norm {\, \cdot \,}_2}$ be the 2-sequence space equipped with Euclidean norm.

Let $c_{00}$ be the space of almost-zero sequences.


Then $c_{00}$ is everywhere dense in $\struct {\ell^2, \norm {\, \cdot \,}_2}$

Proof

Let $\mathbf x = \sequence {x_n}_{n \mathop \in \N} \in \ell^2$.

By definition of $\ell^2$:

$\displaystyle \sum_{i \mathop = 0}^\infty \size {x_i}^2 < \infty$

Let $\displaystyle s_n := \sum_{i \mathop = 0}^n \size {x_i}^2$ be a sequence of partial sums of $\displaystyle s = \sum_{i \mathop = 0}^\infty \size {x_i}^2$.

We have that $s$ is a convergent sequence:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall n \in \N: n > N \implies \size {s_n - s} < \epsilon$

Note that:

\(\ds \size {s_n - s}\) \(=\) \(\ds \size {\sum_{i \mathop = 0}^n \size {x_i}^2 - \sum_{i \mathop = 0}^\infty \size {x_i}^2}\)
\(\ds \) \(=\) \(\ds \size {\sum_{i \mathop = n \mathop + 1}^\infty \size {x_i}^2}\)

Let $\displaystyle N \in \N : \sum_{n \mathop = N + 1}^\infty \size {x_n}^2 < \epsilon^2$

Let $\mathbf y := \tuple {x_1 \ldots, x_N, 0, \ldots}$.

By definition, $\mathbf y \in c_{00}$.

We have that:

\(\ds \norm {\mathbf x - \mathbf y}_2^2\) \(=\) \(\ds \sum_{i \mathop = 0}^\infty \size {x_i - y_i}^2\) Definition of Euclidean Norm
\(\ds \) \(=\) \(\ds \sum_{i \mathop = N + 1}^\infty \size {x_i}^2\)
\(\ds \) \(<\) \(\ds \epsilon^2\)
\(\ds \leadsto \ \ \) \(\ds \norm {\mathbf x - \mathbf y}_2\) \(<\) \(\ds \epsilon\)

By definition, $c_{00}$ is dense in $\ell^2$.

$\blacksquare$


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