In my functional analysis homework problem I was trying to a extend a linear functional from a proper sub-space $Y$ to the whole space $X \supset Y$using the Hahn Banach theorem. In order to do that I needed to find a positive homogenous and sub-linear functional $p$. If I chose it to be $p(x)=\inf_{y\in Y} \| x-y\|$ Now to show it is positive homogenous I need to show that $p(\alpha x)=|\alpha|p(x)$ i.e $\inf_{y\in Y} \|\alpha x-y\|=\inf_{y' \in Y} \|\alpha x-\alpha y'\|=|\alpha|\inf_{y\in Y} \| x-y'\|$
Now $\inf_{y\in Y} \|\alpha x-y\|=\inf_{y' \in Y} \|\alpha x-\alpha y'\|$ holds because we are computing the infimum over the whole subspace Y , in other words $\alpha y'$ is just a scalar multiple of $y$
My reasoning makes sense to me but I do not know how to exactly write it down mathematically
You essentially said $$ p(\alpha x) = \inf_{y \in Y} \|\alpha x - y \| = \inf_{y \in Y} \| \alpha x - \alpha y\| = |\alpha| \inf_{y \in Y} \| x-y\| = |\alpha|p(x), $$ you don't need to say more. Note that $y$ is a parameter so to speak, so you don't need to pick a new symbol $y'$, as long as equality is true you're fine. (In other words, you had it all right.)
Hope that helps,