The quote below is an outline of an intermediate value theorem proof that uses the creeping lemma.

"Assume that $\displaystyle f(x)-L$ is zero for no $\displaystyle c\in[a,b]$. Then, prove that $\displaystyle f(a)-L $ and $\displaystyle f(b)-L$ are of the same sign by applying the creeping lemma with $\displaystyle \rho$ the relation defined by requiring $\displaystyle u\rho\\v$ to be true if and only if $\displaystyle f(u)-L $ and $\displaystyle f(v)-L$ are of the same sign"

Where the creeping lemma is: Let $\displaystyle \rho$ be a transitive relation on the interval $\displaystyle [a, b]$. If each $\displaystyle x\in[a, b]$ has a neighborhood $\displaystyle N_{x}$ such that $\displaystyle u\rho\\v $ whenever $\displaystyle u\in[a, x]\cap\\N_{x}$ and $\displaystyle v\in[x, b]\cap\\N_{x}$ , then $\displaystyle a\rho\\b$.

So, with the above information I need to prove the intermediate value theorem, but I am not sure I understand the outline of the proof entirely and what parts of the lemma I am free to use without proof.

proof: Suppose$\displaystyle f$ is continuous on $\displaystyle [a,b] $ and there is a real number $\displaystyle L$ satisfying $\displaystyle f(a)<L<f(b)$. Now, we can consider this statement in terms of the function $\displaystyle \varphi(x)=f(x)-L$. Suppose there is no $\displaystyle c\in[a,b]$ such that $\displaystyle \varphi(x)=f(x)-L=0$; then, without loss of generality$\displaystyle \varphi(a)<0 $ and $\displaystyle \varphi(b)>0$. So, we want to show, at some point $\displaystyle c\in[a,b]$, $\displaystyle \varphi(c)=0$. I do not know how to use the relation $\displaystyle \rho$ given in the above quote to produce the rest of this proof.

If anyone knows about the creeping lemma and how to apply it, I'd really appreciate some advice on how to move forward.