could use some help answering a question:

Let $A \in \mathscr{B}\left(\ell^{1}\right)$ be defined by $(A f)(m)=\sum\_{n}\left(2^{-m}\left(1-2^{-n}\right) f(n)\right)$. Prove that $A\left(\right.$ ball $\left.\ell^{1}\right)$ is not closed.

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Let $\tau \in \ell^{1}$ be defined by $\tau(n)=2^{-n} .$ Then --***this is where I'm stuck***-- $\tau \in$ $\left(\operatorname{cl} A\left(\right.\right.$ ball $\left.\left.\ell^{1}\right)\right) \backslash A\left(\right.$ ball $\left.\ell^{1}\right) .$ Thus $A\left(\right.$ ball $\left.\ell^{1}\right)$ is not closed.