<s>[INST] If $a,b,c,d,e$ be real number other than $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation $\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$ Question
If $a,b,c,d,e$ be real numbers $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation
$$\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$$

I tried to use Cauchy-Schwarz like
$$\left(\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}\right)(b-a+c-b+d-c+e-d)≥16$$
I think I should prove it using $-2≤a≤b≤c≤d≤e≤2$, but I'm confused about how to do it. Can anyone help me?
 [/INST] $w=b-a, x=c-b,y=d-c, z=e-a$
you have four non-negative numbers $w,x,y,z$ that sum to $e-a\leq 4,$ so
$w+x+y+z \leq 4$   and $\frac{1}{w+x+y+z} \geq \frac{1}{4}$
The harmonic mean of thew is smaller then the arithmetic mean, for
$$ \frac{16}{w+x+y+z}  \leq \frac{1}{w} + \frac{1}{x} + \frac{1}{y} +\frac{1}{z} $$
This book I have refers to an AM-HM inequality for positive real numbers, if there are $n$ numbers
(with equality only when all the numbers are equal)
$$ \left(x_1 + x_2 + \cdots + x_n \right)  \left( \frac{1}{x_1} + \frac{1}{x_2}+ \cdots +\frac{1}{x_n} \right) \geq n^2   $$
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