Ratio Test - Query

Question

I'm trying to understand this solution. Why did we set $ε = r' - r$ and how do we know it's greater than $0$?

Answer 1

We have that $r < r' < 1$, this implies that $0 < r ' - r$ (subtracting $-r$ on both sides of the lef inequality).

Now, since limit $\lim _{n \to \infty} = \left| \frac{a_{n+1}}{a_n}\right| = r$, by definition, we have that, for any $\epsilon > 0$, there exists $N \in \mathbb N$ such that

$$n > N \implies \left| \frac{a_{n+1}}{a_n} - r\right| < \varepsilon $$

In particular, we may choose $\varepsilon = r'- r > 0$. There isn't much else to be said here.