How to demonstrate this inequality

nicoc

New member
Joined
Mar 6, 2019
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Hello, im trying to demonstrate this:

Knowing: [MATH]|a+b| \leq |a|+|b|[/MATH]I have to demonstrate that: [MATH]|a-b| \leq |a-c|+|c-b|[/MATH]
I appreciate if someone can help me how to start.
I dont know if I can do it just with [MATH]| x |[/MATH] properties or if I have to use the following: if [MATH]a \leq b+\epsilon[/MATH], [MATH]\epsilon \gt 0[/MATH] then [MATH]a \leq b[/MATH]
Thanks in advance
 
Are you saying that you are given this fact for a particular pair a, b, or that you are given that this is true for any pair a, b? There is a big difference!

In the latter case, you could replace a with a-c and b with c-b and observe what the inequality becomes. (If that isn't useful, you could try other substitutions.)
 
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