An equivalent condition for positivity

Question

Let $A$ be a $C^*$-algebra. How to prove that $\|1-\frac{x}{\|x\|}\|\leq 1$ implies positivity of $x$? The converse can be shown to be true using functional calculus.

Answer 1

Consider the complex plane and take a point on the unit circle close to $1$. This satisfies your hypothesis but the point need not be positive.