Thread: Inequality in Metric spaces

1. 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.

2. Re: Inequality in Metric spaces

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

3. Re: Inequality in Metric spaces

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 Thank you so much.

4. Re: Inequality in Metric spaces

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.