Number of non-vanishing terms in sum of inner products with compactly supported function A paper (penultimate line in page 7) I'm reading gives the following sum$$\sum_{x\in 2^{-n}\mathbb{Z}^d}|\langle \phi,\psi\rangle|$$
(where $n\in\mathbb{N}$)
and says that, because $\psi$ is a function with compact support, the number of non-zero terms is bounded by $2^{nd}$.
First, I'd imagine what they mean is that the number of non-zero terms in the sum is bounded by some constant multiplied by $2^{nd}$. 
But even so, where does the power of $d$ come from? I'd have said that the number of non-zero terms equals $2^{n}$ multiplied by some element of $\mathbb{Z}^d$, so that we get a constant multiplied by $2^{n}$.

The number of terms is bounded by $2^{n}$ (multiplied by some constant), and for each term there are $d$ components to make a choice for, so the bound we get is $(2^{n})^d = 2^{nd}.$
