Set theory and functions.

Question

Give a one-to-one and onto function $f : [0,1] \to [0,1)$ without using Cantor–Bernstein theorem ?!

I can give $f(x) = x$ and so its one-to-one and onto but its problematic because $f(1)=1$ and so its not in $[0,1)$

Can any one give me hints or help , thanks

Answer 1

Define $f\colon[0,1]\longrightarrow[0,1)$ this way:

• if $x\neq\frac1n$ for each natural $n$, $f(x)=x$;
• $(\forall n\in\mathbb{N}):f\left(\frac1n\right)=\frac1{n+1}$.