Let $c>1$ be a real number, and let $F_{c}$ be the space of sequences $\mathbf{a}=\left(a_{1}, a_{2}, \ldots\right)$ of real numbers $a_{i}$ with $\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ convergent. Show that $\|\mathbf{a}\|_{c}=\sum_{r=1}^{\infty} c^{-r}\left|a_{r}\right|$ defines a norm on $F_{c}$.

Let $F$ denote the space of sequences a with $\left|a_{i}\right|$ bounded; show that $F \subset F_{c}$. If $c^{\prime}>c$, show that the norms on $F$ given by restricting to $F$ the norms $\|\cdot\|_{c}$ on $F_{c}$ and $\|\cdot\|_{c^{\prime}}$ on $F_{c^{\prime}}$ are not Lipschitz equivalent.

By considering sequences of the form $\mathbf{a}^{(n)}=\left(a, a^{2}, \ldots, a^{n}, 0,0, \ldots\right)$ in $F$, for $a$ an appropriate real number, or otherwise, show that $F$ (equipped with the norm $\|.\|_{c}$ ) is not complete.