I am trying to prove the intermediate value theorem stated as:
Let $f:[a,b]\to \mathbb{R}$ be a continuous function with $f(a)=c$ and $f(b)=d$, then $[c,d] \subseteq \operatorname{im}(f)$.
By contradiction via long exact sequence of a pair. So clearly I need to show that if we assume that there exists a point $p\in[c,d]$ that is not in $\operatorname{im}(f)$ we get a contradiction. I think this is a relative homology problem. I'm not sure if I am doing this correctly at all but, we can take the set $U\subseteq \mathbb{R}= \space (-\infty,p)\space \cup \space (p,\infty)$ and we have that $\mathbb{R} \space / \space U$ treats anything not equal to $p$ as the same thing. So now we have the spaces $U$, $\mathbb{R}$, and $\mathbb{R} \space / \space U$. Is this even the right idea so far? And I'm not too sure exactly how to go about creating a long exact sequence from these spaces.
I guess it may just be $$H_0(U)\to H_0(\mathbb{R})\to H_0(\mathbb{R} \space / \space U)\to 0$$
and so $$\operatorname{dim}(\mathbb{R})=\operatorname{dim}(U)+\operatorname{dim}(\mathbb{R} \space / \space U)$$ and we get that $\operatorname{dim}(\mathbb{R} \space / \space U)=-1$ which is a contradiction.