How to prove that the cross product of a countable and uncountable set is uncountable? so my question is, how can you prove that 
  ${\Bbb Z}$ x ${\Bbb R}$ is uncountable?
So far I have tried proving that there is an uncountable subset of ${\Bbb Z}$ x ${\Bbb R}$ without luck and I'm not really sure what I can do
Thanks for any tips you can give!

If $\mathbb{Z}\times\mathbb{R}$ were countable, we could pick a surjection $\pi=(\pi_1, \pi_2)$: $\mathbb{N}\longrightarrow\mathbb{Z}\times\mathbb{R}$. But then $\pi_2$: $\mathbb{N}\longrightarrow\mathbb{R}$ would be a surjection, so $\mathbb{R}$ would be countable.
