= Solution
Put $\rho=r-\sqrt{r^2-1}$. Since $r>1$, $0<\rho<1$ and $\rho^2-2r\rho+1=0$. The nonzero square-summable sequence $\beta_j=\rho^{j-1}$ defines a <continuous linear functional> on $\ell_2$ by
$$
f(\alpha)=\sum_{j\geq1}\alpha_j\rho^{j-1}.
$$
The <Cauchy-Schwarz inequality> proves <continuity>. For every $k$,
$$
f(x_k)=\rho^{k-1}(1-2r\rho+\rho^2)=0.
$$
Hence $f$ vanishes on the <closed linear span> $M$, but $f(e_1)=1$. Therefore
$$
\boxed{M\ne\ell_2.}
$$
More precisely, any vector in the <orthogonal complement> of $M$ satisfies the recurrence $\beta_{k+2}-2r\beta_{k+1}+\beta_k=0$. Its roots are $r\pm\sqrt{r^2-1}$, one less than one and the other greater than one. Square summability eliminates the growing solution, so $M^\perp$ is exactly the span of $(\rho^{j-1})$. Thus $M=\ker f$ is a closed hyperplane of codimension one. This is <annihilator of a second-difference span in l2>.
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