The set of sequences of $\Bbb N$ and $\mathcal{P} (\Bbb N)$ are equinumerous.
My attempt:
Let $S,S_1,S_2$ be the sets of sequences, finite sequences, and infinite sequences of $\Bbb N$ respectively.
Then $S=S_1\cup S_2$ and $\emptyset=S_1\cap S_2$.
We have $|S_2|=|{\Bbb N}^{\Bbb N}|={|\Bbb N|}^{|\Bbb N|}={\aleph_0}^{\aleph_0}$ where $2^{\aleph_0}\le {\aleph_0}^{\aleph_0}\le {\left (2^{\aleph_0}\right )}^{\aleph_0}=2^{(\aleph_0.\aleph_0)}=2^{\aleph_0}$. Thus $|S_2|={\aleph_0}^{\aleph_0}=2^{\aleph_0}$.
Moreover, $|S_1|=\aleph_0 <2^{\aleph_0}=|S_2| \implies |S|=|S_1\cup S_2|=|S_2|$.
Hence $|S|=|S_2|=2^{\aleph_0}=|\mathcal{P} (\Bbb N)|$.
Does this proof look fine or contain gaps and flaws? Thank you for your verification!