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Q:
Conceptual proof of uniqueness of the zero of a locally Lipschitz function
Here is the case of the uniqueness of the zero of the equation $f(x)=0$.
Is there any way to think of the following line of reasoning without using Linear Algebra?
$f$ is locally Lipschitz in some neighborhood of $a\in\mathbb{R}$ so
$
|f(x)-f(a)|=L_f|x-a|
$
$f(a)
eq 0$ by assumption
$f(a)$ is the least element of the set $A=\{f(x)
eq 0\}$
The uniqueness theorem is a corollary of that, but how do we think that $f(a)$ is the least element of $A$ without using Linear Algebra?
A:
You can think of it as an application of the following principle: If $\left \{ f_n \right \}$ and $\left \{ h_n \right \}$ are sequences of real-valued functions defined on a set $X$ and $L$ is a real number such that for each $x \in X$
$\tag{$*)$
$\left \{ n \in \mathbb{N} \ \colon \ f_n(x) \leq L \right \} \subset \left \{ n \in \mathbb{N} \ \colon \ h_n(x) \leq L \right \}$
then $\forall x \in X$ we have $f(x) \leq h(x)$.
So, in your case, since
$A = \left \{ f(x)
eq 0 \right \} = \left \{ f(x) > 0 \right \} \cup \left \{ f(x) = 0 \right \}$
and
$\left \{ n \in \mathbb{N} \ \colon \ f_n(a) \leq L \right \} \subset \left \{ n \in \mathbb{ 0b46394aab
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