  We prove the scale invariant Elliptic Harnack Inequality (EHI) for
non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note
is to provide a simplified self-contained probabilistic proof of EHI in
${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the
Local Central Limit Theorem for simple symmetric random walks on
${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability
function. The uniform Green inequality and the classical Balayage formula then
imply the EHI.
