Inequality in Metric spaces

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

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

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.

Re: Inequality in Metric spaces

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

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

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:

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

you need to prove:

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

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

Re: Inequality in Metric spaces

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Originally Posted by

**Also sprach Zarathustra** To prove:

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

you need to prove:

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

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

I hadn't seen the inequality that way (Headbang) Thank you so much.

Re: Inequality in Metric spaces

Quote:

Originally Posted by

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

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

Start with

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

Again

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

Now add.