# Thread: Bijection from [1, infinity) to R

1. ## Bijection from [1, infinity) to R

The problem says "Find a bijecton $f:[1,\infty)\longrightarrow\mathbb{R}$."

The hint goes on to say you won't be able to give a specific formula. My main question is "Is this even possible?" I don't see how it is.

2. Originally Posted by mathematicalbagpiper
The problem says "Find a bijecton $f:[1,\infty)\longrightarrow\mathbb{R}$."

The hint goes on to say you won't be able to give a specific formula. My main question is "Is this even possible?" I don't see how it is.
Hmm I would do it this way

Map 1 to 3.

Define a bijection between (1,2] and $(-\infty, 2]$ using the usual trick of specifying a point at (2,1) where the number in (1,2] represents an angle in (-pi/2, 0] from the point to the x-axis. (-pi/2 is directly to the left, 0 is straight down).

For non-integers greater than 2, map them to themselves.

For integers n greater than two, map them to n+1.

3. f(x)=(-1)^(⌊x⌋+1)*⌊x/2⌋+(x-⌊x⌋)
is a bijection

It bijectively maps these intervals on each other
[2n-1,2n) -> [n-1,n)
[2n,2n+1) -> [-n,-n+1)
for all n>=1