This is from a discrete math textbook, but obviously calculus-based, so I'm asking it here.

Let be a sequence satisfying

(a) is a positive integer and is a negative integer.

(b) For all , , or .

Prove that there exists , , such that .

I thought about doing proof by contradiction, but I can't work anything out. I do not remember or never came across the regular calculus version of this theorem. Can anyone help?