Arranging numbers in a circle

Here's a question I found from a Chinese book, (which contains no answers), and I found it to be very interesting.

Suppose we have numbers whose sum is 0.

Arrange them in a fixed order in a circular manner.

Show that there exists an integer m, such that , and the sum of any number (1 to n) of consecutive terms starting with is non-negative. (Clockwise direction)

It seems that it is true, after trying with a few numbers.

How do you actually prove it?

Thanks in advance.