I have to prove a property $P(x)$ hold for $\forall x: x \in (0,1]$. I also have a property $F\big(\frac{x}2\big)=F(x)+1$ which is key to prove $P(x)$. If I prove following steps:

$P(x-\epsilon)$ holds for $\forall \epsilon$,where $\frac{x}{2}<x-\epsilon<x$

$P\big(\frac{x}{2}\big)$ holds

$\forall x \in (0,1],\space \space \exists \frac{x}{2} \in (0,1]$.

It this a correct way to prove using induction over real number?