What is the Closest Point to a Compact Set in a Metric Space?

[CarmineCortez]

Homework Statement

Let X be a metric space and let K be any non-empty compact subset of X, and let x be an element of X. Prove that there is a point y is an element of K st d(x,y) leq d(x,k) for every k an element of K.

Homework Equations

triangle inequality

The Attempt at a Solution

i tried proof by contradiction with the triangle inequality, and it didn't get me anywhere

 

[morphism]

Look at the set {d(x,k) : k in K}. Note in particular that it's bounded below.

 

[CarmineCortez]

its bounded below as k gets closer to x is that inf K?