How does this proof show that $\inf_{\alpha \gt 0} \frac{f(\alpha x)}{\alpha}$ is homogenous?

Question

How does this proof show that $\inf_{\alpha \gt 0} \frac{f(\alpha x)}{\alpha}$ is homogenous?

Let $g(x) = \inf_{\alpha \gt 0} \frac{f(\alpha x)}{\alpha}$, $f$ be a convex function, and $t \gt 0$. Then $g$ is homogenous by:

$$g(tx) = \inf \frac{f(\alpha t x)}{\alpha} = t\inf \frac{f(\alpha t x)}{t\alpha} = tg(x) = t \inf \frac{f(\alpha x)}{\alpha}$$

But then it must be the case that

$$\inf_{\alpha} \frac{f(\alpha t x)}{t\alpha} = \inf_{\alpha} \frac{f(\alpha x)}{\alpha}$$

Why is this the case?

Answer 1

$$\inf_{\alpha>0 }\frac{f(\alpha tx)}{t\alpha }=\inf_{t^{-1}u>0}\frac{f(ux)}{u}=\inf_{u>0}\frac{f(ux)}{u},$$ where the last equality comes from the fact that $t^{-1}u>0\iff u>0$.