closest point compact set

[CarmineCortez]

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

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.

2. Relevant equations

triangle inequality


3. The attempt at a solution

i tried proof by contradiction with the triangle inequality, and it didn't get me anywhere
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution

[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?