Q1) Assuming that |R|=|[0,1]| is true, how can we rigorously prove that |$\displaystyle R^2$|=|[0,1] x [0,1]|? How to define the bijection?

Q2)

**Prove that |[0,1] x [0,1]| ≤ |[0,1]|** __Proof:__ Represent points in [0,1] x [0,1] as infinite decimals

x=$\displaystyle 0.a_1a_2a_3...$

y=$\displaystyle 0.b_1b_2b_3...$

Define f(x,y)=$\displaystyle 0.a_1b_1a_2b_2a_3b_3...$

To avoid ambiguity, for any number that has two decimal representations, choose the one with a string of 9's.

f: [0,1] x [0,1] -> [0,1] is one-to-one, but not onto.

This one-to-one map proves that |[0,1] x [0,1]| ≤ |[0,1]|.

Now

**how can we formally prove that f is a one-to-one map (i.e. f(m)=f(n) => m=n)? **All textbooks are avoiding this step, they just say it's obviously one-to-one, but this is exactly where I'm having trouble. How to prove formally?

Thanks a million!

[also under discussion in math links forum]