In the model proof below for the Intermediate Value Theorem it is written that $L_p=\{y\in[a,b]\quad\text{such that}\quad f(y)\lt{q}\}$. Then $c\in{L_q}$ but $d\notin{L_q}$. Shouldn't it be that $a\in{L_q}$ but $b\notin{L_q}$ because $L_q\subseteq{[a,b]}$ and not $L_q\subseteq{[c,d]}$? I keep picking up on parts of my lecture notes that seem wrong to me.
Yes i mean this is probably a typo. $L_q$ is in the domain, but c and d are in the range. You are 100% correct.