Fractional and non-fractional Hardy inequality on a lattice ${\mathbb{Z}}^d$

We present simple proofs of a discrete fractional and non-fractional Hardy inequality, Our constants are explicit, but not optimal. In the class of power weights, we get a complete picture of when the non-fractional Hardy inequality holds, for any dimension $d$ of the lattice ${\mathbb{Z}}^d$ and exponent $0<p<\mathrm{\infty }$.