Let the space $l^2$ be equipped with its usual metric , $d_2$. Let $x ∈ l^2$ and consider the following sequence $(z_n)^∞_{n=1}$, where
$z_1= x = (x_1,x_2,x_3,...)$
$z_2 = (x_1/2,x_2/2,x_3/2,...)$
$z_3 = (x_1/3,x_2/3,x_3/3,...)$
and so on. Is the sequence Cauchy and convergent?
$d(z_n,z_m) = \sqrt{\sum_{i=1}^\infty\ (x_i/n-x_i/m)^2}$ $ = \sqrt{\sum_{i=1}^\infty\frac{x_i(m-n)^2}{(nm)^2}}$ $\to 0$ as n, m $\to \infty$
Therefore it is Cauchy
$d(z_n,0) = \sqrt{\sum_{i=1}^\infty\ (x_i/n)^2}$ $\to 0$ as n $\to \infty$
So it is convergent
is this correct?