How to prove: $\sum_{i=1}^{k}|x_{i+1}-x_{i}| \geq \max_{i \in [k+1]}x_{i} - \min_{i \in [k+1]}x_{i}$?

Question

How to prove:

$$\sum_{i=1}^{k}|x_{i+1}-x_{i}| \geq \max_{i \in [k+1]}x_{i} - \min_{i \in [k+1]}x_{i}\quad ? $$

Is it even correct? I know how to prove that $$\sum_{i=1}^{k}|x_{i+1}-x_{i}| \geq x_{k+1}- x_{1}$$ Note that $\forall x_{i}\geq 0$.

Answer 1

Let $1\leqslant a\lt b\leqslant k$. Then $$ \sum_{i=1}^{k}|x_{i+1}-x_{i}| \geqslant\sum_{i=a}^{b-1}|x_{i+1}-x_{i}| \geqslant \left|\sum_{i=a}^{b-1}x_{i+1}-x_{i}\right|=\left| x_{b}-x_{a}\right|. $$ This implies that for all $1\leqslant a,b\leqslant k$, $$ \sum_{i=1}^{k}|x_{i+1}-x_{i}| \geqslant x_b-x_a. $$ Can you conclude?