Closeness of a set in a infinite dimensional Hibert Space and projection of $0$

Question

I began to read Clarke's Nonsmooth Analysis and the first exercise in chapter is

Let $X$ be a real Hilbert space that admits a countable orthonormal basis $\{e_i\}_{i=1}^\infty$ and set $$ S := \left\{ \frac{1+i}{i}e_i : i\geq1 \right\}. $$ Prove that $S$ is closed, and that $\text{proj}_S(0) = \emptyset$.

1. For the closeness, $S_k:=\left\{ \frac{1+i}{i}e_i : 1\leq i\leq k \right\}$ with $1\leq k $ is closed (since it is a discrete set) then $S=\bigcup_{k\geq1}S_k$ is closed.

Now, for the projection (defined by the set of points that the distance of $0$ to $S$ is minimal) , how can this set be empty ? I mean, shouldn't it exist a point in $S$ that the distance to $0$ is the smallest then every other point in $S$?