Let's give a bijection between $ \{ x \in \mathbb{R}: 2 \ge x \ge 1 \} $ and $\mathbb{R} $

Question

Can I maybe use in some way Hilbert's Grand Hotel Paradox or in which way can I find such a bijection?

Do you want an explicit bijection or just proof of a bijection? — Bill O'Haran, Apr 18 '18 at 12:05
I would like to find an explicit bijection, the exercise is using this notation $ 2 \ge x \ge 1 $ — Zauberkerl, Apr 18 '18 at 12:06

score 1 · Answer 1 · answered Apr 18 '18 at 12:13

1

$\Bbb{R}$ bijects with $(0,1)$ via a scaled and translated $\tan$ function, so you're effectively looking for a bijection between $[0,1]$ and $(0,1)$. See Bijection between an open and a closed interval.

answered Apr 18 '18 at 12:13

Teddy38

3,309

could you maybe give me some more hints? – Zauberkerl Apr 18 '18 at 12:52
Pick two sequences one from $0$ toward $1/2$ (increasing) and the other from $1$ toward $1/2$ (decreasing) neither sequence containing $1/2.$ Then on these "shift toward $1/2$ by one step", and use the identity map on the rest. – coffeemath Apr 19 '18 at 01:13

Let's give a bijection between $ \{ x \in \mathbb{R}: 2 \ge x \ge 1 \} $ and $\mathbb{R} $

1 Answers1