Approximation of Lipschitz functions by finite sums

Question

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a Lipschitz function. Is it true that we can approximate $f$ by a finite sum as follows: for all $x\in \mathbb{R}$ $$ f(x)=\sum_{i=1}^nb_i|x-a_i|,\quad a_i,b_i\in \mathbb{R}? $$ If yes, could you please provide a hint on how to prove it? Thanks.

Answer 1

The function $\sin x$ is Lipschitz on $\mathbb R.$ Suppose

$$\sin x= \sum_{k=1}^{n}b_k|x-a_n|$$

for $x\in\mathbb R.$ For $x\ge \max \{a_1,\dots,a_n\},$ we then have

$$\sin x= \sum_{k=1}^{n}b_k(x-a_k) = \left(\sum_{k=1}^{n}b_k\right)x-\sum_{k=1}^{n}b_ka_k.$$

The function on the right is linear. This says $\sin x$ is eventually linear, contradiction.