Number of images from $\mathbb{N}$ to {0, 1}. 
Are the number of images from $\mathbb{N}$ to {0, 1} countably infinite or uncountably infinite?


I was thinking of counting in base 2 to make a bijection between  $\mathbb{N}$ and {0, 1}. So, a table like this one:
 Image  |  0  1  2  3  ...         <- Here is N
-------------------------------
  0     |  0  0  0  0  ...         <- And here is the image in {0,1}
  1     |  1  0  0  0  ...
  2     |  0  1  0  0  ...
  3     |  1  1  0  0  ...
  4     |  0  0  1  0  ...
...

So, right now, in the left column, I can identify every number of $\mathbb{N}$ with an image on the right. Which makes it countably infinite, by definition.
But a co-student, claimed that the prof said it was uncountably infinite. Any idea why? 

You can use that if $A\subseteq \mathbb{N}$ then $f(a)=1$ if $a\in A$ and $f(b)=0$ if $b\notin A$ then $\mid \{0,1\}^\mathbb{N}\mid=\mid P(\mathbb{N})\mid$ by Cantor's theorem, $\mid \mathbb{N}\mid<\mid P(\mathbb{N})\mid$.
