# Distance between point and set

1. Aug 10, 2013

### saxen

\in1. The problem statement, all variables and given/known data
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

2. Relevant equations

3. 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!

2. Aug 10, 2013

### Staff: Mentor

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.

3. Aug 10, 2013

### 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.

4. Aug 10, 2013

### Staff: Mentor

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

5. Aug 12, 2013

### saxen

Thank you! I got it right.