If $\displaystyle a,b \in N $ with $\displaystyle a/b$ {i.e: a divides b} .prove that $\displaystyle a \leq b $
Well $\displaystyle b=aq$ for some $\displaystyle q \in \mathbb N$ (here we assume that $\displaystyle 0 \notin \mathbb N$ or else the statement is false - take $\displaystyle b=0$ and $\displaystyle a>0$). Then $\displaystyle q\geq 1$ so $\displaystyle b=aq>a\times 1 =a$.