Why is $\limsup_{n\rightarrow \infty} (1/n) \log(a_n) < 0 \Rightarrow \sum_{n=0}^{\infty} a_n < \infty$ true?

Why is $\limsup_{n\rightarrow \infty} (1/n) \log(a_n) < 0 \Rightarrow
\sum_{n=0}^{\infty} a_n < \infty$ true?

Let $(a_n)$ be a real values series.
Why is the following implication true? $$\limsup_{n\rightarrow \infty}
(1/n) \log(a_n) < 0 \Rightarrow \sum_{n=0}^{\infty} a_n < \infty$$