# Construct an injective function that takes a function $\mathbb{N} \to \mathbb{R}$ and outputs an injective function $\mathbb{N} \to \mathbb{R}$

How can I build an injective function that takes any function from natural to real numbers and gives back an injective function from natural to real numbers??

Thanks... EDIT: How can I prove that for a set A={Al of the ingective functios from natural to real numbers} the equality |A|=א IS TRUE??

• Must there by any additional relationship between the arbitrary function $f: \mathbb{N}\to\mathbb{R}$ and its image, the injective $\bar f: \mathbb{N}\to \mathbb{R}$? EDIT: Just realized that the mapping $\bar (-)$ must be injective. – Thorsten Jan 2 '15 at 10:44
• There are easier ways to show that the sets of all functions from the naturals to the reals and the set of all injective functions from the naturals to the reals have the same cardinality. So if that's the objective, maybe rephrase the question? – Henno Brandsma Jan 2 '15 at 10:44

Pick an injective function $g\colon\mathbb R\to[0,1)$ and for $f\colon \mathbb N\to\mathbb R$ define $\phi(f)\colon \mathbb N\to\mathbb R$ by $\phi(f)(n)=n+g(f(n))$. (This works also if we replace $\mathbb N$ with $\mathbb Z$)
$$\big(\varphi(f)\big)(n)=\sum_{k\le n}2^{f(n)}$$
This one does not work with $\Bbb N$ replaced by $\Bbb Z$, however.