Understanding proof from paper - if sequence of traces is unbounded then $E$ is not in the spectrum

Question

I am reading a paper by Suto - The spectrum of a quasiperiodic Schrodinger operator. I am having trouble with understanding the proof of one of the lemmas, which seems to be a simple linear algebra argument. I'd be happy if someone could try to explain it to me - from what I understand it does not really require understanding the paper, just this particular linear algebra argument. I already understand the most part, but there is one argument I don't understand - I'll emphasize where that is.

Here is a bit of background. We are given some discrete Schrodinger operator $H$ on $\ell^2(\mathbb Z)$. The author also sometimes thinks about it as a simple difference operator, which allows to apply $H$ to general sequences - not just to $\psi\in\ell^2(\mathbb Z)$ (this is a slight abuse of notation, but never mind that).

For fixed $E\in\mathbb R$, The author defines a sequence of matrices $(M(n))_{n=1}^\infty$ with some nice property, and then defines the sequence of traces $x_n:=\frac{1}{2}tr(M(n))$, and prove the following:

If $E$ is not in the spectrum of $H$, then the sequence $x_n$ is unbounded

The idea of the proof is as follows:

1. He proves that if the sequence $x_n$ is bounded ($|x_n|\leq M$), then any solution of $(H-E)\psi=0$ (not necessarily in $\ell^2$) satisfies a nice algebraic identity - let's call it $(*)$.

2. He says the following (this is exactly where I lose him - I emphasize the relevant part):

"If $E$ is not in the spectrum of $H$, then there exists a unique solution $\psi\in \ell^2(\mathbb Z)$ to the equations $(H-E)\psi(k)=\delta_{0k}$. For $k\neq 0$, these are homogeneous, hence $\psi$ satisfies (*).""

3. The rest of the proof easily follows from $(*)$.

The thing I really don't understand is this homogeneity argument. The equation $(H-E)\psi(k)=\delta_{0k}$ is not homogeneous. Although "most" of the components of the vector $(H-E)\psi$ are equal to $0$, one cannot say that $(H-E)\psi=0$, which is required in order to apply $(*)$.

Can anyone explain to me how the idea behind this? How can you use a property of solutions to the homogeneous equation, although your vector is not zero in all of its components? I can give more details about the paper if you wish - I tried to not confuse with too much information, and I think that the answer is less about the specific details.

Thanks in advance!