let $(a_n)$ be the sequence $(1, 0, 2, 0, 3, 0, . . .)$. Find $\lim \inf$, $\lim \sup$ of $(a_n)$ and $(b_n)$ where $b_n= (a_1+a_2+...+a_n)/n$

Question

Finding $\lim \inf$ and $\lim \sup$ of $a_n$ is easy. Without using any theorems one can also see $\lim \sup$ of $b_n$ is $+\infty$ as every even term is of form $\frac{n+2}{8}$ but how does one find $\lim \inf b_n$? Even if I were to use cauchy 1st theorem on limits $\lim \inf b_n$ should be between $0$ and $\infty$ , but isnt that already clear without the theorem . What approach should I adopt? Any idea would be helpful. Thanks.