Consider (x,y) in R2 with x,y both in (0,1). Write x as some decimal $x=0 .a_1a_2a_3...$ and $y=0.b_1b_2b_3b_4...$
Now, write z in R as $a_1b_1a_2b_2a_3b_3...$.
If y and x are both finite, pad the one of lower length with zeros until they are of the same length. If x is finite and y is infinite, pad x with infinite zeros.
This is a bijection because each x, y maps uniquely onto some x in R, and from each x one can derive x,y.