The cardinal number

Question

Let $c$ be the cardinal number of $[0,1]$, i.e. $|[0,1]|=c$. Notice that $|A|\cdot|B| = |A\times B|$ and $|\mathbb{R}| = c$. Prove that $c\cdot c=c$. Don't use $ab=\max\{a,b\}$ where $a,b$ are infinite.

Answer 1

Here's an approach you can take. For every $x\in(0,1],$ show that there is a unique sequence $a_1,a_2,a_3,\dots$ such that every $a_n\in\{0,1\},$ there are infinitely-many $a_n=1,$ and $$x=\sum_{n=0}^\infty\frac{a_n}{2^n}.$$ Use this to show that the set $S$ of all such sequences together with the sequence of all $0$s, has the same cardinality as $[0,1].$ Then, given an ordered pair of two sequences of $S$--say $a_1,a_2,a_3,\dots$ and $b_1,b_2,b_3,\dots$--we create another such sequence by interleaving them as $a_1,b_1,a_2,b_2,\dots.$ Use this to show that $S\times S$ is in bijection with $S,$ whence $[0,1]\times[0,1]$ is in bijection with $[0,1].$