In this paper, Lorentz claimed that the set $\mathcal F$ of all almost convergent sequences is closed and non-separable in the normed linear space $l_\infty$ with respect to the sup-norm and I've understood the proof in my another question.
Lorentz also claimed that $\mathcal F$ is nowhere dense in $l_\infty$.
My Question : How can I show that $\mathcal F$ is nowhere dense in $l_\infty?$
Best Answer
Since $\mathcal{F}$ is closed, it is nowhere dense in $\ell^\infty$ if and only if it has empty interior in $\ell^\infty$.
Suppose not.
Then there is $x \in \mathcal{F}$ and $\varepsilon > 0$ such that for $y \in \ell^\infty$, $\|x-y\|_\infty \leq \varepsilon$ implies that $y \in \mathcal{F}$. Take arbitrary $z \in \ell^\infty \setminus \mathcal{F}$ with $\|z\|_\infty = 1$. Then consider $y = x + \varepsilon z$. Since $\|x-y\|_\infty = \varepsilon \|z\|_\infty = \varepsilon$, we must have $y \in \mathcal{F}$. But $\mathcal{F}$ is a linear space so we then get that $z \in \mathcal{F}$ which is a contradiction.
Note that the above is really a proof of the fact that if $Y$ is a closed proper subspace of a normed space $X$ then $Y$ is nowhere dense in $X$.