The set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable? [duplicate]

How can I prove that the set of all functions from $\mathbb{N} \to \{0, 1\}$ is uncountable?

Edit: This answer came to mind. Is it correct?

This answer just came to mind. By contradiction suppose the set is $\{f_n\}_{n \in \mathbb{N}}$. Define the function $f: \mathbb{N} \to \{0,1\}$ by $f(n) \ne f_n(n)$. Then $f \notin\{f_n\}_{n \in \mathbb{N}}$.

Solution 1:

Hint: Show that $\{0,1\}^\mathbb N$ is equinumerous with $\mathcal P(\mathbb N)$ and use Cantor's theorem to conclude there is no bijection between $\mathbb N$ and $\mathcal P(\mathbb N)$.

Solution 2:

You can proof it by contraposition. I will identify the functions with sequence so $a_{n}$=$a(n)$. Now lets say it's countable, now let $a_{nk}=a_k(n)$ be the $k$-th function. Now we construct the function $$b(k)=\left\{ \begin{array}{rl} 1 & a_{kk}=0\\ 0& a_{kk}=1\end{array}\right.$$ You should be able to do the rest.