**Question** Does anyone know of a good reference which I can cite for the *finite* dimensional version of Mountain Pass Lemma?


**Motivation** I am writing a paper and found myself using the following result:



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> Let $f$ be a proper smooth real-valued function on $\mathbf{R}^3$ such that $f(0) = 0$, $f|\_{B\_1(0)} \geq 0$, $f|\_{\partial B\_1(0)} \geq 1$ and $\exists p \in {\partial B\_2(0)}$ such that $f(p) = 0$. Then $\exists q\in \mathbf{R}^3 \setminus B\_1(0)$ such that $f'(q) = 0$ and $f(q) \geq 1$. 
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For the time being I referred to Ambrosetti and Rabinowitz's JFA article for the mountain pass lemma, but citing a Banach space version for a finite-dimensional Euclidean space application gives me a funny feeling. (Also, if feels like such a result could in principle be found in not-so-advanced undergraduate textbooks...)



Finite dimensional "Mountain Pass Lemma"
My book *An Invitation to Morse Theory*, 2nd Edition, Springer Verlag 2011 describes the finite dimensional Mountain Pass Lemma in Example 2.53. There I work on a compact manifold, but the compactness of the manifold can be substituted by a properness assumption on the function. In the same section I explain a more general Min-Max principle (Thm. 2.51) and in Example 2.53 I explain how this implies the Mountain Pass Lemma.


