Regarding Lemma 4, the $J$’s are compact subintervals, what are they a subinterval of? I am just trying to understand that part while watching this lecture, I know it has nothing to do with the proof, but for my knowledge I was curious.

Map $f:I\to I$ is a continuous map of the interval $I$ to itself. $J$ is a compact subinterval of $I$, say
$J=[a,b] \subseteq I$ with $a < b$.
Dr. Abdulla

Dr. Abdulla,

Regarding Lemma 4, the $J$’s are compact subintervals, what are they a subinterval of? I am just trying to understand that part while watching this lecture, I know it has nothing to do with the proof, but for my knowledge I was curious.

Thanks

Kaleb,

Map $f:I\to I$ is a continuous map of the interval $I$ to itself. $J$ is a compact subinterval of $I$, say

$J=[a,b] \subseteq I$ with $a < b$. Dr. Abdulla

Thank you sir.