Abstract: Let $H$ be a one-dimensional discrete Schrödinger 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 $H_0 + \tfrac{1}{4} V^2$ has at least one bound state then the same is true for $H_0 +V$. Further, if $H_0 + \tfrac{1}{4} V^2$ has infinitely many bound states then so does $H_0 +V$. Consequences include the fact that if $\liminf_{|n|\to\infty} |nV(n)| \geq 1$, $H_0 +V$ has infinitely many bound states; the signs of $V$ are irrelevant. Some extensions to higher dimensions are also discussed. This is joint work with David Damanik, Rowan Killip, and Barry Simon.