Let $T\colon \ell_\infty \rightarrow \ell_\infty : (a_1,a_2,\ldots) \mapsto (a_1,a_2/2,a_3/3,a_4/4,\ldots) $.
Show that $\operatorname{range}(T)$ is not dense in $\ell_\infty$. I want to ask for a hint or a solution to this problem.
Let $T\colon \ell_\infty \rightarrow \ell_\infty : (a_1,a_2,\ldots) \mapsto (a_1,a_2/2,a_3/3,a_4/4,\ldots) $.
Show that $\operatorname{range}(T)$ is not dense in $\ell_\infty$. I want to ask for a hint or a solution to this problem.
Hint: If $(x_n)$ is in the range of $T$, then $x_n\rightarrow 0$. What is a lower bound of the distance in $\ell_\infty$ from the vector $(1,1,\ldots)\in\ell_\infty$ to such an $(x_n)$?