(related to Corollary: \(0x=0\))

- Let $x$ be a real number.
- We want to show that the multiplication of $x$ by zero results in zero, or $0\cdot x=0.$
- From the existence of zero, it follows that $0= 0 +0.$
- Multiplying this equation by $x$ results by the distributivity law in $0x=0x + 0x.$
- On the other hand we have $0x=0x + 0,$ applying the existence of zero once again.
- Because of the uniqueness of zero, we can compare the last two equations.
- This comparison results in \(0x=0\) for all \(x\in\mathbb R\).∎

Parts: 1

**Forster Otto**: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983