Uniqueness of real numbers represented as products of integer powers of primes

Question

Let $p_k$ represent the $(k+1)$th prime number. My hypothesis is that all positive real numbers may be represented as some infinite product $$\prod_{k=0}^\infty p_k^{e_k}$$ (where $e_k \in \mathbb{Z}$); and moreover that this product is unique. Intuitively I am certain this is true, but I cannot imagine how I could go about proving it.

EDIT: some rational examples for clarity:

$1 = 2^0\cdot3^0\cdot5^0\cdot7^0\cdot11^0\cdot\dots$

$\frac{22}{7} = 2^1\cdot3^0\cdot5^2\cdot7^{-1}\cdot11^1\cdot\dots$

$100 = 2^2\cdot3^0\cdot5^2\cdot7^0\cdot11^0\cdot\dots$

Answer 1

An infinite product $\prod x_i$ of real (and also complex) numbers converges only if the sequence $(x_i)$ converges to 1. In your case you have to have $p_i^{e_i} \to 1$. Which is only possible if $(e_i)$ is eventually vanishing.

Answer 2

No.

The product will not converge unless almost all exponents are zero. In other words, you get exactly the positive rational numbers.