  Let $H$ be a one-dimensional discrete Schr\"odinger operator. We prove that
if $\sigma_{\ess} (H)\subset [-2,2]$, then $H-H_0$ is compact and
$\sigma_{\ess}(H)=[-2,2]$. We also prove that if $H_0 + \frac14 V^2$ has at
least one bound state, then the same is true for $H_0 +V$. Further, if $H_0 +
\frac14 V^2$ has infinitely many bound states, then so does $H_0 +V$.
Consequences include the fact that for decaying potential $V$ with
$\liminf_{|n|\to\infty} |nV(n)| > 1$, $H_0 +V$ has infinitely many bound
states; the signs of $V$ are irrelevant. Higher-dimensional analogues are also
discussed.
