show $1/x$ does not equal $ 0$

Question

Suppose that $x\neq 0$, show that $\frac{1}{x}\neq 0$

This seems like it should be simple, but I don't even know how to start to prove this.

Answer 1

If $\frac1x=0$ and $x\ne0$, then $x\cdot\frac1x=x\cdot 0$, id est $1=0$.

Answer 2

If $x\ne0$ then $x>0$, in which case $\frac1x>0$ (otherwise $x\frac1x=1<0$),

or $x<0$, in which case $\frac1x<0$ (otherwise $x\frac1x=1<0$).

In either case, $\frac1x\ne0$.