Let $(M,\,g,\,d,\,vol_g)$ be a Riemannian manifold with metric $g$, geodesic distance $d$ and volume form measure $vol_g = \sqrt{\det(g_{ij})}\cdot m$ ($m$ = Lebesgue measure) and a Lipschitz map $f:K\to f(K) \subseteq M$, with Lipschitz constant $k$, where $K\subseteq M$ is compact. What I would like to show is that for any Borel set $A\subseteq K$ one has $$vol_g(f(A)) \leq Ck^n vol_g(A)$$ where $C>0$ is a constant (in my source of the problem the author claims that $C=1$, but for my purpose it suffices to be any constant).
What I've managed so far is to use the regularity of measure and the well-known Vitali's Covering Lemma to reduce the problem in showing that for every ball $B_x(r)$, where $x\in K$ and $r$ is as small as we would want it to be, we must have $$vol_g(f(B_x(r))) \leq C k^n vol_g(B_x(r)).$$
Using the Lipschitz continuity we can get to $$vol_g(f(B_x(r))) \leq k^n vol_g(B_{f(x)}(r))$$ and this is where I'm stuck. I'm thinking of using Bishop-Gromov and symbolic manipulation but, firstly, I don't know anything about a lower Ricci bound and, secondly, even if I did it seems like an overkill, since the author describes the above problem exactly as in my title and gives not even a hint about its proof.
How should I move on?