Justification for inequality

Suppose $(M,g)$ is a Riemannian manifold with $\text{diam } M \leq D$. Let $u : M \rightarrow [0,\infty]$ be smooth. Define $R_i := 2^{-i}D$ so that $B_i := B(x,R_i)$ for a fixed $x \in M$. In particular, $u(x) = \lim_i\frac{1}{|B_i|}\int_{B_i}u(y)dy$ by the Lebesgue differentiation theorem. Notice that $M = B_0$. Define $u_{B_i} = \frac{1}{|B_i|}\int_{B_i}u(y)dy$.

I am not seeing how to derive the following inequality: $$|u(x)-u_{B_0}| \leq \sum_{i=0}^\infty |u_{B_i}-u_{B_{i+1}}|.$$