Question

In: Advanced Math

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there...

Let S ⊆ R be a nonempty compact set and p ∈ R. Prove that there exists a point x_0 ∈ S which is “closest” to p. That is, prove that there exists x0 ∈ S such that |x_0 − p| is minimal.

Expert Solution

Consider the function defined by . We claim that this is continuous. Let be arbitrary; then for any we claim that

Notice that if we prove this claim then will be proven to be (uniformly) continuous in , by -definition of continuity.

Now, by triangle inequality we have

and similarly,

This shows that

which proves

Now, because is continuous on compact set , it attains a minimum in . Thus, there is such that

This completes the proof.

Colby Messinger answered 1 year ago

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