Consider the Hilbert space $X=l^2$, equipped with the usual norm $‖·‖_2$
Let the subspace $Y\subset l^2$ consist of sequences with only finitely many nonzero terms, i.e.,
$Y= \big\{y=(y_k)^∞_{k=1} \in l^2:~ y_k \neq 0 \text{ for finitely many } k\big\}$.
Let $x\in l^2$ be defined as $x= (1,1/2,1/3,1/4,...)$
(i) If $y_m\in l^2$ is defined as
$y_m= (1,\frac{1}{2},\frac{1}{3},...\frac{1}{m},0,...)$,
compute $p_m=‖x−y_m‖_2$ as an infinite series $(p_m)^∞_{m=1}$.
(ii) Hence, determine $d(x,Y) = \inf_{y\in Y}‖x−y‖_2$.
For (i), I have the infinite series $p_m = (0,..,0,\frac{1}{m+1},\frac{1}{m+2},...)$
Unsure of how to compute part ii from this
For each $m\in\mathbb{N}$, $$ p_m=\sqrt{\sum_{n=m+1}^\infty\frac1{n^2}}. $$ So, $\lim_{m\to\infty}p_m=0$. Since $p_m$ is the distance from $x$ to some element of $Y$, it follows from this that $d(x,Y)=0$.