Establishing an inequality between the first term of an infinite geometric series and the infinte sum?

Question

An infinite geometric series has the first term a and sum to infinity b, where b $\neq 0$. Prove that a lies between 0 and 2b.

$ \rightarrow \text{Since the series converges, } r \text{ has to be between 0 and 1 }\\ \text{(}\text{using the geometric series formula, i.e } \frac{a(1 - r^n)}{1 - r}\text{):}\\ \text{The sum}=b=\frac{a}{1-r}\text{, where } r \text{ is the common ratio.}\\ \rightarrow b - br = a $

Ok. Now what? I'm stuck.

Answer 1

Hint:

Note that $-1<r<1$. Therefore, $0<1-r<2$.

$a/b=1-r$