Recurrence Relation, a question about the relation between $A_n$ and $A_{n+1}$

Question

Given the following recurrence relation:

$$A_{n+1} = {1 \over 4(1-A_n) }$$ $$ A_1 = 0 $$ Can I safely assume that if- $$\forall n \in \mathbb{N}, \ A_n < 1/2$$ then

$$\forall n \in \mathbb{N}, \ A_{n+1} < {1\over4(1-{1\over2})} ={1\over2}$$

Thus determining that $$\forall n \in \mathbb{N}, \ A_n < 1/2$$

by induction.

or am I missing a step?

Answer 1

if $A_n<\frac{1}{2}$ we get $4A_n<2$ and this is equivalent to $2<4-4A_n$ and this is equivalent to $\frac{1}{4(1-A_n)}<\frac{1}{2}$ and this is equivalent to $A_{n+1}<\frac{1}{2}$