Claim: $R^\infty \text{ is not a Banach space when equipped with its natural product topology}$ I need help proving this 'obvious' claim. I just got acquainted with a definition of a product topology and the concept does not seem to be easy to work with. How would I even go about showing whether $R^\infty$ is metrizable?
Edit: Intuitively, I know that $d(x,y)=\sum_{j=1}^{\infty}\frac{1}{2^j} \frac{|x_j-y_j|}{1+|x_j-y_j|}$ should prove that $R^{\infty}$ is metrizable.