Prove that between two zeros of there is a zeros of
First observe that:
Now if and are consecutive zeros of then the signs of and would be different (the other possibility that one or both of these is zero does not occur as that would imply that for one ,or both, of the zeros that , but such a point cannot be a zero of ).
But the sign of and being different, implies that is positive at one root and negative at the other, so as is continuous, by the intermediate value theorem must be zero at some point between and . which proves the result.