Originally Posted by

**Plato** For the last question look back at the set of sequences I suggested.

Here are the first three terms: $\displaystyle x_1 = \left( {1,0,0,0, \cdots } \right),\quad x_2 = \left( {0,1,0,0,0 \cdots } \right),\quad x_3 = \left( {0,0,1,0,0, \cdots } \right)$.

Is it clear to you that $\displaystyle d\left( {x_1 ,x_2 } \right) = 1,\quad d\left( {x_1 ,x_3 } \right) = 1,\quad d\left( {x_3 ,x_2 } \right) = 1$?

Every term in that sequence is one unit from any other term in the set.

But that is a bounded, infinite set. Can it have a limit point?