  The discrete Schr\"odinger operator with the Dirichlet boundary condition is
considered on the half-line lattice $n\in \{1,2,3,\dots\}.$ It is assumed that
the potential belongs to class $\mathcal A_b,$ i.e. it is real valued, vanishes
when $n>b$ with $b$ being a fixed positive integer, and is nonzero at $n=b.$
The proof is provided to show that the corresponding number of bound states,
$N,$ must satisfy the inequality $0\le N\le b.$ It is shown that for each fixed
nonnegative integer $k$ in the set $\{0,1,2,\dots,b\},$ there exist infinitely
many potentials in class $\mathcal A_b$ for which the corresponding
Schr\"odinger operator has exactly $k$ bound states. Some auxiliary results are
presented to relate the number of bound states to the number of real resonances
associated with the corresponding Schr\"odinger operator. The theory presented
is illustrated with some explicit examples.
