Definition: A sequence $(x_n)$ is bounded if there exists a number $M \gt 0$ such that $ \left| x_{n} \right| \leq M$ for all $n \in \mathbb{N}$.
Question: Is there any reason why one can't replace $ \left| x_{n} \right| \leq M$ with the strict inequality $ \left| x_{n} \right| < M$? Since $M$ is a fixed number it seems that $1$ could always be added to make the inequality strict, but maybe I am missing something... If I am right, and the inequality can always be strict, then why didn't they write it that way in the first place?