TITLE: An alternative proof of the Łojasiewicz inequality
QUESTION [11 upvotes]: Is there a "brute force proof" of the Łojasiewicz inequality? By "brute force" I mean a proof without introducing the machinery of semianalytic sets and so on but only using elementary results (i.e., standard Calculus 1+2). I admit i hadn't time to think about it properly, I thought I could prove it first for homogeneous polynomials (and I got stuck) and then extend the result to analytic functions, but I could be wrong... To avoid misunderstandings I recall here the statement of the Łojasiewicz inequality.
Let $f\in C^{\omega}(B_{1},\mathbb{R})$ (real analytic function on the open unit ball centered at the origin) then there are $\alpha\in (0,1)$, $C>0$ and $\varepsilon$ sufficiently small such that  $\forall x\in B_{\varepsilon}$ the following holds true:
$$|f(x)-f(0)|^{\alpha}\leq C|\nabla f(x)|.$$
Thank you in advance.

REPLY [2 votes]: No, there is no elementary proof of that result even for polynomials of 2 variables. The proof of this result an be reduced to a more elementary one (the case of monomials) by using the theory of sigularities, but this is not at all elementary, cf. the recent works of Paul Feehan.