Three problems from G.Rosenstein "Linear orderings" (from the end of Chapter 2 and beginning of Chapter 4):
1) Is there a nondecreasing function from irrationals onto reals?
2) Is there a nondecreasing function from reals onto irrationals?
3) Is there an increasing function from reals into irrationals? (In other words, are reals a subordering of irrationals?)
Any hints would be appreciated.
(Please tag the question set-theory order-theory)
Best Answer
Here is a simple proof of (3), without continued fractions.
For any real number x, let f(x) be the real obtained by interleaving the binary digits of x with the binary digits of pi, a fixed irrational number. This is clearly order-preserving, and f(x) is irrational since the digits will not repeat. QED