Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$ Prove that $[0,1]$ isn't homeomorphic to $\mathbb{R}$
My first thought is that there can not be a continuous bijection, $f$, from $[0,1]$ to $\mathbb{R}$ because a continuous function that maps $[0,1]\rightarrow \mathbb{R}$ must be bounded so there can't be a surjection. So that would then be the proof. 
Though I believe I am incorrect in my thinking because a hint for the assignment says to use the intermediate value theorem. That is "Suppose $f:[a,b]\rightarrow\mathbb{R}$ is continuous. If $f(a)<\delta<f(b)$ or $f(b)<\delta<f(a)$ then $\delta=f(c)$ for some $c\in[a,b]$". 
Why is my first thought wrong and how is the IVT useful?

You can remove two points from $[0,1]$ without disconnecting it. That's not true of $\mathbb{R}$.
Your first thought wasn't wrong, and I don't know what the hint is getting at. This is just another proof, demonstrating that multiple proofs exist.
