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Proof

(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 \(x\in\mathbb R\).
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References

Bibliography

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