Distance between point and set

[saxen]

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Homework Statement

Denote by d(x,A) = inf |x-y|,y \in A, the distance between a point x \in R^n and a set A \subseteq R^n. Show

|d(x,A)-d(z,A)| \leq |x-z|

In particular, x → d(x,A) is continuous

Homework Equations

The Attempt at a Solution

I have no idea on how to prove this. I drew a picture and the result seemed intuitive but I don't know how to prove it mathematically.

Appriciate any help!

 

[mfb]

For closed sets, this is easy to show with the triangle inequality. For general sets, I would try to apply the same argument for a converging series of elements of A.

 

[saxen]

Ah yes! I actually tried the triangle inequality but failed. I am going to try again!

Could you please elaborate some more on the second part? I have been stuck on similar questions because I do not understand this argument.

 

[mfb]

If there is no y in A such that d(x,A)=d(x,y), there is a sequence y_i such that d(x,y_i) converges to d(x,A) (for i->infinity).

 

[saxen]

Thank you! I got it right.