Prove property of $A$ implies $\inf A \le 2$

Question

Set $A$ is a proper subset of reals and has a property that for every $a \in A$ there exists $b \in A$ such that $a \ge 2b - 2$. How to prove that $\inf A \le 2$?

Answer 1

Hint:

If $2b-2\leq a$ then $b\leq \frac{a+2}{2}$ (so $b$ is less than or equal the average between $a$ and $2$).

Moreover, if $a>2$ then $\frac{a+2}{2}<a$.

We conclude that if $a>2$ then there exists $b\in A$ such that $b<a$ can you finish now?