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.