Find a Cauchy sequence that does not converge

Question

I am supposed to look at $l_0$, the set of all sequences with finitely many non-real elements in $(l_0,d_{\infty})$. It is just that I don't quite understand how the $d_\infty$-metric is defined on sequences(and basically else, it would be nice if anyone could elaborate...).

If you look at the sequence $\{x_n\} = (\frac{1}{n},0,0,....,)$, this is clearly a Cauchy-sequence, is it then so that the supremum of all the numbers in the sequence will converge to 1?

EDIT: In my notes, i have defined $d_{\infty}(x,y) = \max_{1 \leq j \leq n} |x_j - y_j|$

Answer 1

Try the sequence $x_n = (1, 1/2, 1/3, 1/4, \dots, 1/n, 0, 0, 0, \dots )$.