Here is how I proved that set of all numbers from N -> {0,1} is uncountably infinite. It is similar to how cantor proved that real numbers are countably infinite.
Let us assume there are countably infinite functions from N -> {0,1}. So every natural number is mapped to either 0 or 1.
That is, functions can be represented as
f1 : 010010...
f2: 1011101...
f3: 0001011...
.
. and so on
Note that the 0 1 sequence here is how the natural numbers are being mapped to 0 or 1 for each function. Now if someone claims that this countably infinite, I can pick the 1st digit from 1st function, 2nd digit from 2nd function, 3rd digit from 3rd function and so on and flip those digits and I will have a function which is not in this supposedly countably infinite list. Thus, I can say that the claim that all set of functions from N -> {0,1} is countably infinite is false. Therefore, this is uncountably infinite. Its easier to understand this proof if you already understand cantor's proof of how real numbers are uncountable.