convergence equivalence of lim inf and lim sup

Let $(a_n)_{n \in\ \mathbb{N}},~(b_n)_{n \in\ \mathbb{N}}$ sequences of positive real numbers, so that $\alpha = \lim\limits_{n \rightarrow \infty} \inf \frac {a_n}{b_n}$ and $\beta = \lim\limits_{n \rightarrow \infty} \sup \frac {a_n}{b_n}$ exists with $\alpha, \beta >0$.

Show that: $\sum\limits_{n=0}^{\infty}a_n$ converges $\Leftrightarrow$ $\sum\limits_{n=0}^{\infty}b_n$ converges.

My idea is to use the comparison test, but i don't really know how to start..

1 Answer

$$\forall \varepsilon > 0, \exists N \in \Bbb N, \forall n \ge N, \alpha - \varepsilon \le \cfrac{a_n}{b_n}\le \beta + \varepsilon$$