  We study 1D discrete Schr\"odinger operators $H$ with integer-valued
potential and show that, $(i)$, invertibility (in fact, even just Fredholmness)
of $H$ always implies invertibility of its half-line compression $H_+$ (zero
Dirichlet boundary condition, i.e. matrix truncation). In particular, the
Dirichlet eigenvalues avoid zero -- and all other integers. We use this result
to conclude that, $(ii)$, the finite section method (approximate inversion via
finite and growing matrix truncations) is applicable to $H$ as soon as $H$ is
invertible. The same holds for $H_+$.
