Consider the Hilbert space $\mathcal{l}^2$. I have to show that the span of $$S:=\{e_1 - e_2, e_2 - e_3, \cdots \}$$ is dense in $\mathcal{l}^2$.
My idea is to show that something is dense in $\mathcal{l}^2$ if and only if $e_1$ (or more generally $e_i$) is in the closure of $\mathbb{R}S$. As it turns out $e_1$ is not in the closure as one can easily check. So is the question wrong or is there a problem in my understanding.
Your statement that $e_1$ is not in the closure of the span of $S$ is incorrect. In fact, we have: $$ x_n := (e_1 - e_2) + \left(1 - \frac{1}{n}\right) (e_2 - e_3) + \left(1 - \frac{2}{n}\right) (e_3 - e_4) + \cdots + \frac{1}{n} (e_n - e_{n+1}) = \\ \left(1, -\frac{1}{n}, \ldots, -\frac{1}{n}, 0, \ldots\right). $$ (Here, the last term has $n$ copies of $-\frac{1}{n}$.) However, then it is easy to calculate that $\lVert e_1 - x_n \rVert_2 = \frac{1}{\sqrt{n}}$, and each $x_n$ is in the span of $S$; therefore, $e_1 \in \overline{\operatorname{span}(S)}$.
