Criteria for convergence of series in terms of asymptotic equivalence of general terms.

Question

I'm trying to prove the following: Let $(a_n)_{\geq0}$ and $(b_n)_{n\geq0}$ be sequences of nonnegative real numbers convergent to $0$. If $a_n\sim b_n$ when $n\to\infty$ (i.e. $\lim_{n\to\infty}\frac{a_n}{b_n}=1$), is it then true that $$ \sum_{n=0}^\infty a_n<\infty\Leftrightarrow\sum_{n=0}^\infty b_n<\infty.\ \ ? $$

I've trying to prove this for a while but haven't got any luck yet and so I started doubting if it really holds. Can someone please help me here?

Answer 1

Yes. It is true.

Hint. If $a_n\sim b_n$ then for $0<\epsilon<1$ there is $N$ such that for $n\geq N$, $1-\epsilon<a_n/b_n\leq 1+\epsilon$. Therefore $$(1-\epsilon)\sum_{n=N}^{\infty} b_n\leq \sum_{n=N}^{\infty} a_n\leq (1+\epsilon)\sum_{n=N}^{\infty} b_n.$$