Differences between finite supremum and bounded supremum?

Question

Are there any differences between, $$\sup_{n\geq 1} a_n<+\infty$$ and $$\sup_{n\geq 1} a_n \leq M \text{ for some } M\geq 0 \text{ independent of } n$$ here $(a_n)_{n\geq 1}$ is some given positive sequence.

Thank you very much if you could provide a detailed clarification!

Answer 1

There are no differences. Both expressions mean:

the sequence $(a_n)$ is bounded from above.