Bijection between $\mathbb{Z}\times\mathbb{Z}\times\dots$ and $\mathbb{R}$

Question

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?

Answer 1

First note that as joriki commented there are many uncountable sets which are not of the same size as the real numbers. There might be uncountable sets which are strictly smaller than the real numbers.

To both your questions, however, the answer is yes. Note that:

$$\mathbb{Z\times Z\times Z\times\dots = Z^N}\\\mathbb{R\times R\times R\times\dots = R^N}$$

Now we have this: $$|\mathbb R|=2^{|\mathbb N|}\leq|\mathbb Z|^{|\mathbb N|}\leq|\mathbb R|^{|\mathbb N|}=2^{|\mathbb{N\times N}|}=2^{|\mathbb N|}=|\mathbb R|$$