elementary set theory - Bijection between $mathbb{Z}timesmathbb{Z}timesdots$ and $mathbb{R}$

Is there a bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ for countably infinitely many $\mathbb{Z}$'s and $\mathbb{R}$? That is, is $\mathbb{Z}\times\mathbb{Z}\times\dots$, repeated countably infinitely many times, uncountable?