# Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2.

We now define for each $k \geq 2$ the function $f_k : M \rightarrow \mathbb{R}$ as follows:

$$f_k((a_i)_i) = \prod_{i\geq 1} \frac{a_i^k}{a_i^k -1}$$

It is fairly obvious, that $f_k$ takes values between 1 and 2 for all $k$ and sequences $(a_i)_i$.

Consider now the combined function $f: M \rightarrow \mathbb{R}^\mathbb{N}$ given by $$f(a) = (f_2(a),f_3(a),f_4(a),\ldots)$$

My question is now: Is it true or wrong if I claim that $f$ injective?

• I suspect that the answer is "true". Presumably you can actually identify $a_1$ by noting that when $k$ is very large, $(1-f_k((a_i))^{-1})^{-1/k}$ is really close to $a_1$. Then you can remove the $a_1$ factor and identify $a_2$ similarly, and so on. Sep 14, 2018 at 18:51