Define Intermediate-Value theorem
Using this theorem, show that there is a root of P(x)=x^3+x^2+x-1 in the closed interval [0,1].
P(x) is continuous. Since P(0)=-1 and P(1)=2 there must be a number in such that (Since 0 is between -1 and 2). We can do even better. We have
Therefore there is a root between ...
This defines Intermediate-Value theorem and uses it in a proof.