TITLE: Definition of a bounded sequence
QUESTION [12 upvotes]: My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$  
But then what about the sequence $(0, 0, ...)$? In that case, can't $M$ be $0$?  
Wikipedia provides the following definition, which seems more reasonable to me:
A sequence $\{x_n \}$ is said to be bounded if $\exists M \in \mathbb R$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$  
Is my professor's definition inaccurate or imprecise in any way?

REPLY [3 votes]: Both your Prof. and wikipedia is right infact your Prof. is more precisely telling you where that $M$ lies.
The only difference between both the definitions is that your Prof. take $M\gt 0$ and wikipedia is taking $M\in \mathbb R$ but don't you think wikipedia definition also indirectly says that $M\ge 0$ because $|x_n| \ge0 ,$thus it can't be made smaller than a negative number. And $M=0 \Rightarrow |x_n| \le 0 \Rightarrow x_n=0$ and if sequence is constant zero then any $M\gt 0$ will also work.