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Alternative proof of $a\times0= 0$
I was trying to find a proof of $a\times0 = 0$ by myself (assuming commutativity, associativity, distributivity, etc) and I came up with $$ a+0=a(1) \implies 1 = \frac{a+0}{a} = \frac aa + \frac 0a = ...
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$a > b+1 \Rightarrow a>x>b$?
If I have $a,b \in \mathbb R$ such that $$a > b+1 $$
It is assured that $\exists\space x \in \mathbb Z: a>x>b$
Does this property have some special name?
How can this be proved?
This idea ...
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