this is one of the problems I found in a book I'm studying:
In a Hilbert space $\mathbb{H}$, find a complete and linearly independent sequence $(x_k)$ which is not a Schauder basis.
With it, there is a hint that reads: contruct $x_k$ so that it converges to some nonzero vector in $\mathbb{H}$; show that such sequences are never Schauder bases.
I have no idea where to start. I tried taking any $x\in\mathbb{H}\setminus\{0\}$ and taking a sequence which converges to $x$, but then what? I can't even guarantee that such a sequence will be complete and linearly independent (in fact, I can't even imagine how it can be linearly independent, since it will eventually get arbitrarily close to $x$). Any help is appreciated, thanks.
Edit: we say that a sequence $(x_k)\subset\mathbb{X}$ is complete if $\forall x\in\mathbb{X}$ and $\forall\varepsilon>0$ there exist $a_k\in\mathbb{R}$ such that $||x-\sum_{k=1}^n a_kx_k||\leq\varepsilon$. The coefficients $a_k=a_k(\varepsilon)$ may depend on $\varepsilon$ in this case.