TITLE: Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?
QUESTION [5 upvotes]: Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?
My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all $n\ge N$.  And we know $0< a_n$ for all $n\in \mathbb{N}$, particularly $0<a_n$ for all $n\ge N$. Maybe by contradiction: suppose that $L<0$, then $L<0<a_n$ for all $n\in \mathbb{N}$, particularly for all $n\ge N$. Then $0<-L<a_n-L$ for all $n\in \mathbb{N}$, particularly for all $n\ge N$. It follows: for all $\epsilon >0$, there is a $N\in \mathbb{N}$ such that $0<|-L|=-L<|a_n-L|<\epsilon$ for all $n\ge N$, which can't be true.
Is my proof ok?

REPLY [2 votes]: Here is a proof that I believe is more succinct than all of the above. 
Suppose $\lim a_n = a <0$ . 
Let $\epsilon=-a>0$ .
By hypothesis, there exists an $n$ large enough such that $\left|a_n-a\right|<\epsilon =-a$ 
$\Rightarrow a_n-a<-a $
$\Rightarrow a_n <0$ 
Contradiction.