# Inequality in Metric spaces

• 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 $w,x,y,z \in X$:
$|d(w,x)-d(y,z)|\le d(w,y)+d(x,z)$

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