I started going through the book Understanding Analysis and kinda don't understand the section about cardinality in the first chapter; imo it leaves a lot of important stuff that is necessary for the text to convey before jumping into the excersises.
The major argument that supports this notion is with the problem 1.5.4 (c) part where we are told to prove that $[0,1)$ has the same cardinality as $(0,1)$. The proof is quite creative and beautiful but even with the right ideas that I had, I wasn't able to prove it due to an additional "trick". Basically we do a specific case of the "Hilbert's Hotel" experiment where we construct a function which "pushes" $f(0)$ by construction a set (let the set be represented as $S$) such that the set is an infinite, countable set which only has arbitrary elements of $[0,1)$ and most importantly $0$. The function is$$f(x)=\begin{cases}x_{n+1}&(x_{n} \in S),\\ x&(x \in [0,1)/S).\end{cases}$$I don't know if I am the one to blame but the "execution" of the creativity has become a little arduous for me. Before this section every thing was very clear but now I think that the author does, sometimes, miss the vital information.
After this painfully boring rant, I wish to ask you:
what else should I go through and practice before this, or am I just worrying too much and should take it slow?
Any help would be appreciated.