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

Half-line compressions and finite sections of discrete Schr\"odinger operators with integer-valued potentials