I have the following lemma that I would like to find a source to cite for. Let $L$ be a subset of $\mathbb Z^d_{>0}$. I would like to claim that the limit $$\lim_{z \to (1,\ldots,1)^-} (\sum_{v \in L} z^v) \prod_{i=1}^d(1-z_i)$$ exists. Here $z^v$ mean $z_1^{v_1}z_2^{v_2}\ldots z_d^{v_d}$, with $v=(v_1,\ldots, v_d)$ a vector in $\mathbb Z^d$.

For the 1-dimensional case (d=1), an argument using Cesaro sum (http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation) will show that the limit exists, and I believe it can be extended to higher dimension as well, though it can be very messy, and I would like to avoid that. Does anyone know a cleaner method (or even better, a source to cite directly from) to solve this problem? Thanks in advance.