TITLE: Sum of all real number for any interval.
QUESTION [5 upvotes]: We know that sum of natural numbers over any interval always exists.
For example sum from 0 to 10 of all natural numbers is $$S=\sum_{n=0}^{10}{n}=\frac{0+10}{2}\times{10}=55$$ 
But what about real numbers?
Is sum of all real numbers from 0 to 10 infinite?
If it is infinite how to prove it?

REPLY [3 votes]: $$
\sum_{n \atop {\vphantom{\Huge A}x_{n}\ \in\ \left[0,1\right]}}x_{n} >
\sum_{n = 1}^{\infty}{1 \over n}
$$