Intuitive proof of the Intermediate value theorem to first year calculus students I want to give an informal proof of the intermediate value theorem to my calculus students. They have just learned about continuity and they don't know yet about derivatives among more other advanced topics.
Do you think it's possible?
 A: Assume that $f(a)<c<f(b)$, bisect the interval $[a,b]$ into $[a_{2},b_{2}]$ such that $f(a_{2})<c<f(b_{2})$ (if this fails, that means we have reached the goal). Proceed this again to get $[a_{3},b_{3}]$... so the left endpoints and right endpoints converge to a point $\eta$, continuity then implies $f(\eta)=c$. 
A: First year was four years ago for me, and I remember being rather overwhelmed by analysis the first time I met it. Having said that, I wasn't a particularly remarkable student, and it also depends on where it is that you're teaching.
My point is, it depends on what you want your informal proof to include. I'm pretty convinced you could provide a formal proof and they'd be able to understand - it may just take a bit of hard work. Perhaps they don't have the tools they need for it?
An informal proof could be anything from drawing a line between two points, to providing a reasonable sketch of the proper proof. I think an intuitive understanding of the intermediate value theorem isn't difficult for first years, so perhaps you can strive for a reasonable sketch proof.
