Inequality in Metric spaces

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January 7th 2012, 07:27 PM
MATHNEM

Inequality in Metric spaces

Let $(X,d)$ be a metric space. Prove that for all $w,x,y,z \in X$ :

$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

I haven't been able to prove it, I'm sure the triangle inequality should be used but I don't know how. Thanks in advance.
January 7th 2012, 10:10 PM
Also sprach Zarathustra

Re: Inequality in Metric spaces

Quote:

Originally Posted by MATHNEM

Let $(X,d)$ be a metric space. Prove that for all $w,x,y,z \in X$ :

$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

I haven't been able to prove it, I'm sure the triangle inequality should be used but I don't know how. Thanks in advance.

To prove:

$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

you need to prove:

1. $d(w,x)-d(y,z) \le d(w,y)+d(x,z)$

2. $d(w,x)-d(y,z)\geq -(d(w,y)+d(x,z))$
January 7th 2012, 10:42 PM
MATHNEM

Re: Inequality in Metric spaces

Quote:

Originally Posted by Also sprach Zarathustra

To prove:

$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

you need to prove:

1. $d(w,x)-d(y,z) \le d(w,y)+d(x,z)$

2. $d(w,x)-d(y,z)\geq -(d(w,y)+d(x,z))$

I hadn't seen the inequality that way (Headbang) Thank you so much.
January 8th 2012, 02:43 AM
Plato

Re: Inequality in Metric spaces

Quote:

Originally Posted by MATHNEM

Let $(X,d)$ be a metric space. Prove that for all $w,x,y,z \in X$ :
$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

Start with
$\begin{align*}d(w,x) &\le d(w,y)+d(y,x)\\ d(w,x)-d(x,y) &\le d(w,y) \end{align*}$

Again
$\begin{align*}d(x,y) &\le d(x,z)+d(z,y)\\ d(x,y)-d(z,y) &\le d(x,z) \end{align*}$ .

Now add.

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