TITLE: Proving a set is uncountable.
QUESTION [7 upvotes]: I need to prove that the set of all functions $\mathcal{F}:\mathbb{N}\rightarrow \left \{ 0,1 \right \}$ is uncountable.
I'm not too sure at all how to do this. My initial idea was to try and show that $\mathcal{F} \rightarrow \mathbb{N}$ was not a bijection but I'm not clear at all and need some big help.
Thanks!!!

REPLY [4 votes]: I will NOT make this a proof by contradiction.
Let $f_1,f_2,f_3,\ldots$ be any sequence of such sequences.  Thus $f_1$ is the sequence $f_1(1),f_1(2),f_1(3),\ldots$.
Try to show that the sequence $(1-f_k(k))_{k=1}^\infty$ is not one of the sequences $f_1,f_2,f_3,\ldots$ although is is a member of $\mathcal F$.

REPLY [2 votes]: You should try a diagonalisation argument, like Cantor's.
Suppose there were a complete countable list of functions, then it should be easy to give a construction of a new function that differs with each.