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which are within the same interval is equal or less to the difference of the marginal values of the interval?

I have the following inequalities: $a \le x \le b$ $a \le y \le b$

How to prove that: $|x - y| \le b - a $

I tried subtracting y from the first inequality to get:

$a - y \le x - y \le b - y$ but don't know how to conclude: $| x - y | \le b - a$ ?

2 Answers 2

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Hint: Multiply the second inequality with $-1$ and then add the two. Can you continue this reasoning? Also note that $\left|x\right|\le y\iff -y\le x\le y$ for $y\ge 0$

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    @Shirohige You are 100% correct.2012-12-28
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Hint Prove that

$x-y \leq b-a \, \mbox{and} \, y-x \leq b-a \,.$