The excerpt below is taken from Rosenthal's A First Look at Rigorous Probability. $K$ refers to the cantor set.
My question refers to the statement "It is easily checked that $f(K) =[0,1]$. I am thinking that this can by proved by taking any number in $[0,1]$, writing the binary expansion for it (that is, write it of the form $\sum_{n=1}^\infty b_n\cdot2^{-n},\ b_n\in \{0,1\}$ and then show that there is a point in the cantor set that will give $d_n = b_n \forall n$. How would I do this last step? That is, how would I show that such a point exists in the cantor set, $K$?
To state the question again: how I would show that there is a $y\in K$ which corresponds to some point in $[0,1]$?
A secondary question is: Can we show that $f(K) = [0,1]$, where $f$ is as in the attached image, without explicitly using binary/ternary expansions, and preferably also not using compactness? It is not important that this secondary question is answered: if the first question is answered and this one is not, I will select an answer for the question and then probably just create a separate question eventually for this and add a bounty if necessary.
Thank you.
Note: This question is technically addressed here, but the answers seems to say to me "here is a function, it is a surjection, without explaining why it is a surjection.