Let we have the following linear operator on a normed space $$T:\ell^2 \to \ell^2$$ $$x=(a_1,a_2,a_3,\ldots) \to Tx= \left(\frac{a_1} 1,\frac{a_2} 2,\frac{a_3} 3, \ldots \right)$$ Let $R(T)$ the range of the operator $T$
Prove the set $R(T)$ is dense by using the definition of density
Let $b=(b_1,b_2,b_3,\ldots)\in\ell^2$ and assume $\varepsilon>0.$ The problem is to prove that some member of the range of $T$ is within a distance $\varepsilon$ of $b=(b_1,b_2,b_3,\ldots).$ I'd look at $a = (a_1,a_2,a_3,\ldots) = (b_1, 2b_2, 3b_3,\ldots,nb_n,0,0,0,\ldots).$ Make $n$ big enough to make the distance between $a$ and smaller than $\varepsilon.$ Making only finitely many terms nonzero is what assures you that $a\in\ell^2.$