Simple Proof From Loomis and Sternberg's Calculus

[Bashyboy]

Claim: $x \vec{0} = \vec{0}$

Let $x$ be some arbitrary scalar in $\mathbb{R}$.

$x \vec{0} = x ( \vec{0} + \vec{0}) \iff$

$x \vec{0} = x \vec{0} + x \vec{0} \iff$

$x \vec{0} - x \vec{0} = x \vec{0} + x \vec{0} - x \vec{0} \iff$

$\vec{0} = x \vec{0}$

Because we had supposed x was arbitrary, meaning that we have not assumed anything about x beyond the fact that it is merely in $\mathbb{R}$, then this statement must be true of all real numbers.

 

[BruceW]

yeah! very nice. ok, I am now fully convinced that $a\mathbf{v}=\mathbf{0}$ implies either $a=0$ or $\mathbf{v}=\mathbf{0}$ :) It takes quite a lot of work to prove things that seem fairly intuitive. But it's kind of satisfying too, right? I'm more physics than maths, so I don't have much experience doing proofs. But I would like to learn a bit more 'proper mathematics'. In fact, I was reading about groups this week.