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If $X$ is a metric space and $x_0\in X$. Let $x$ and $x'$ be any points of $X$. I want to unerstand why the following inequality is correct:

$d(x,x_0)-d(x',x_0) \leq d(x,x')$

I understand that if we break it down, we have:

$d(x,x_0) \leq d(x,x')+d(x',x_0)$

$d(x',x_0) \leq d(x,x')+d(x,x_0)$

So I see that

$d(x,x_0)-d(x',x_0) \leq d((x',x_0)-d(x,x_0)$ but I still can't see why:

$d(x,x_0)-d(x',x_0) \leq d(x,x')$

Joaquin Liniado
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aribaldi
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1 Answers1

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Just take the $d(x,x_0) \leq d(x,x') + d(x',x_0)$ and simply subtract $d(x',x_0)$ from both sides.

SquirtleSquad
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