ThmDex – An index of mathematical definitions, results, and conjectures.
Result R5168 on D993: Real number
Infinite product of real numbers in right-closed unit interval vanishes iff infinite sum of duals diverges
Formulation 0
Let $x_0, x_1, x_2, \ldots \in (0, 1]$ each be a D993: Real number.
Then \begin{equation} \prod_{n = 0}^{\infty} x_n = 0 \quad \iff \quad \sum_{n = 0}^{\infty} (1 - x_n) = \infty \end{equation}
Formulation 1
Let $x_0, x_1, x_2, \ldots \in (0, 1]$ each be a D993: Real number.
Then \begin{equation} x_0 x_1 x_2 x_3 \cdots = 0 \quad \iff \quad (1 - x_0) + (1 - x_1) + (1 - x_2) + (1 - x_3) + \cdots = \infty \end{equation}
Formulation 2
Let $x_0, x_1, x_2, \ldots \in (0, 1]$ each be a D993: Real number.
Then \begin{equation} \lim_{N \to \infty} \prod_{n = 0}^N x_n = 0 \quad \iff \quad \lim_{N \to \infty} \sum_{n = 0}^N (1 - x_n) = \infty \end{equation}
Proofs
Proof 0
Let $x_0, x_1, x_2, \ldots \in (0, 1]$ each be a D993: Real number.