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\noindent
\parbox{2.25in}{\bf Math 4710/5710-101}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Oct. $22^\mathrm{nd}$, 2014}

\vspace{0.4in}

Recall that $\mathbb{R}^\omega = \mathbb{R} \times \mathbb{R} \times \cdots$ is the set of sequences of real numbers: ${\bf x} = (x_1,x_2,x_3,\dots)$.

\vspace{0.05in}

Our default topology on $\mathbb{R}^\omega$ is the product topology. However, it can also be given the box topology or the uniform topology. 

\vspace{0.05in}

Recall that the {\it uniform topology} is the topology induced from the uniform metric. The {\it uniform metric} on $\mathbb{R}^\omega$ is the function $\bar{\rho}({\bf x},{\bf y}) = \mathrm{sup} \{ \bar{d}(x_i,y_i) \;|\; i=1,2,\dots \}$ where ${\bf x} = (x_1,x_2,\dots)$, ${\bf y} = (y_1,y_2,\dots)$, and $\bar{d}(a,b) = \min \{ |a-b|, 1\}$. 

\vspace{0.05in}

Let $\mathbb{R}^\infty = \{ {\bf x} \in \mathbb{R}^\omega \;|\; {\bf x} \mbox{ is eventually zero } \}$. So ${\bf x} =(x_1,x_2,\dots) \in \mathbb{R}^\infty$ implies that there exists some $N \in \mathbb{Z}_{>0}$ such that $x_n=0$ for all $n \geq N$. So in other words, the coordinates of ${\bf x}$ are non-zero only finitely many times.

\vspace{0.05in}

{\sc The Problem}: Determine the closure of $\mathbb{R}^\infty$ in $\mathbb{R}^\omega$ in the product, box, and uniform topologies.





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