Backward orthogonalisation of a given set

Hello,
I have a question, that states that an orthogonal set $\{u_1,u_2,\ldots\}$ is called the backward orthogonalisation of the set $\{x_1,x_2,\ldots\}$ in a Hilbert space, if $\overline{\text{Sp}}\{u_n,u_{n+1},\ldots\} = \overline{\text{Sp}}\{x_n,x_{n+1},\ldots\}$ for all n.
I am requested to find the backward orthogonalisation of the set $\{x_1,\ldots\}$ , where $x_n=e_n+e_{k+1}$ , where $\{e_n\}$ is the standard basis of $l_2$ .

I am pretty sure I should be able to express $u_k$ in the form $x_k+\sum_{j=k+1}^\infty a_jx_j$ for all k, but that quickly leads me to $a_n=\pm 1$ for all n, and that can't be it.
Do you have any idea?

Thanks.