TITLE: Trying to embed the natural numbers in $[0,1]$
QUESTION [0 upvotes]: This seems to be very easy but I don't see it. How can I embed $\mathbb N$ in $[0,1]$ with a function? I have a function $\phi:\mathbb N\rightarrow\mathbb R$ with $\phi(n) = \beta^n$ for $0<\beta<1$. I want to write down a function $\varphi:[0,1]\rightarrow\mathbb R$ that basically does the same as $\phi$ (i.e. has the same graph but continuous).
To be more formal: I have to find a function $\varphi$ mapping $[0,1]$ to $\mathbb R$ such that for every $n\in\mathbb N$ there exists a $x\in[0,1]$ such that $\varphi(x) = \phi(n)$.

REPLY [1 votes]: Take the funcion $f:\mathbb{N}\longrightarrow \mathbb{R}$ such that $f(n)=\frac{n+1}{n+2}$. This function preserves the order and its image is isomorphic to $\mathbb{N}$.
{"set_name": "stack_exchange", "score": 0, "question_id": 3716996}