How can I explicitly construct a bijective function from [0,∞) to (0,∞)? I see that [0,∞) – N = (0,∞) – Z+, but what do I do next?
Also, can anyone show me how to do this by explicitly constructing a bijective function from [0,1] to [0,1) and bijection function from [0,1] to (0,1)?
Thanks!!
Hints. For the first question, suppose you left all the non-integers alone. That leaves just the non-negative integers, which you now have to map onto the positive integers. Can you find a good way to do that? That is, can you figure out what to put in place of the question mark in
$$ f(x) = \begin{cases} x & x \text{ is not an integer} \\ ? & x \text{ is an integer} \end{cases} $$
For the second question, you can apply the same kind of trick, you just need to find a countably infinite sequence of numbers in $[0, 1]$ that begins with $1$. For instance, $1, 9/10, 8/10, 7/10, \ldots$ would work if you only needed a sequence of length $10$. But you need a sequence of infinite length that stays inside the interval $[0, 1]$. Can you figure one out?
