calculus - I'm trying to show that $limlimits_{x rightarrow infty}frac{1}{x}=0$, is my argument ok?

Here is my try:

We know that $\lim\limits_{y\rightarrow 0} y = 0$, so define $y=\frac{1}{x}$, then:

$\lim\limits_{x \rightarrow \infty}\frac{1}{x} = \lim\limits_{y\rightarrow 0}\frac{1}{\frac{1}{y}}= \lim\limits_{y \rightarrow 0} 1 \frac{y}{1}=\lim\limits_{y \rightarrow 0} y = 0$

Is my argument correct? I don't know if it's right or wrong.

If it's ok, how do I justify each step?

If it's wrong, which step is illegal?

Thanks.