What if $a_n = o(1)$ and $a_n = O(1)$?

Question

What happens when $a_n = o(1)$ and $a_n = O(1)$? What can we say about $a_n$ and its growth rate?

Answer 1

$a_n = o(1)$ means that $(a_n)$ converges to $0$.

$a_n = O(1)$ means that $(a_n)$ is bounded.

So any $o(1)$ is also a $O(1)$.

Answer 2

As already noticed

$$a_n=o(1) \implies a_n=O(1)$$

Indeed recall that

$$a_n=o(1) \iff a_n=1\cdot \omega(n) \quad \omega(n)\to 0$$

therefore

$$\lim_{n\to \infty} \frac{a_n}1 =\lim_{n\to \infty} \frac{1\cdot \omega(n)}1 \to 0$$