Does the equality $\sigma(cX)=\sigma(X)$ holds where $c\neq 0$ and $x$ is r.v?

Question

I think the answer is yes because if $c>0$ then

$\sigma(cX)=\sigma\{(cX\leq x)|x\in \mathbb{R}\}=\sigma\{(X\leq \frac{x}{c})|\frac{x}{c}\in\mathbb{R}\}=\sigma(X)$

and in the same manner if $c<0$ then

$\sigma(cX)=\sigma\{(cX\leq x)|x\in \mathbb{R}\}=\sigma\{(X\geq \frac{x}{c})|\frac{x}{c}\in\mathbb{R}\}=\sigma(X)$

Is my reasoning correct?

Edit: $\sigma(X)$ is the minimal $\sigma$-algebra which makes the function $X$ a random variable