Thanks, that was very helpful. It makes sense that I need to be thinking of a sequence of sequences. I was just having trouble wrapping my head around this L space.
And you are of course right about my typo in the problem set-up, I fixed it.
I was attempting to find a counterexample to the problem below. I think I may have, but was ultimately left with more questions than answers.
Consider the space, L, of all bounded sequences with the metric \rho_1
\displaystyle \rho_1(x,y)=\sum\limits_{t=1}^{\infty}2^{-t}|x_t-y_t|
Show that a...