This proof looks sound to me, but it seems too simple to work.
Proof of intermediate value theorem:
Suppose $t \in [m,M]$, with $m$ and $M$ the minimum and maximum values of $f$ on $[a,b]$, respectively, and $f(c) \neq t$ for any $c\in [a,b]$. By the extreme value theorem, $ m \leq f(c) \leq M$ $ \forall c \in [a,b]$. Therefore $f(c) \in [m,M] $ $\forall c \in [a,b].$ Since $t \in [m,M]$, there is a $c \in [a,b]$ such that $f(c) = t$.