Cantor's Proof of (0,1]~[0,1] Bijection Explained

[catherinenanc]

1. Can anybody elxplain to me (or point me to a URL of an explanation) how Cantor proved the existence of a bijection (0,1]~[0,1]?
2. It's not for homework. I have to understand it generally for a paper I am writing.
3. I think it has something to do with transfinites but I can't get it.

 

[LCKurtz]

I don't know if it is how Cantor did it but it is easy enough. To map [0,1] to (0,1] try this:
$0\rightarrow \frac 1 2$
$\frac 1 2 \rightarrow \frac 1 3$
$\frac 1 3 \rightarrow \frac 1 4$
$\frac 1 4 \rightarrow \frac 1 5$
...
$\frac 1 n \rightarrow \frac 1 {n+1}$
Map all other points into themselves. That gives a 1-1 correspondence between the two intervals.