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.

Solutions

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

Next > < Previous

Related Solutions

Please prove that: A nonempty compact set S of real numbers has a largest element (called...

Please prove that: A nonempty compact set S of real numbers has a largest element (called the maximum) and a smallest element (called the minimum). By the way, I think a minimum is provided by -max(-S)

Let A ⊂ R be a nonempty discrete set a. Show that A is at most...

Let A ⊂ R be a nonempty discrete set a. Show that A is at most countable b. Let f: A →R be any function, and let p ∈ A be any point. Show that f is continuous at p

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N...

Let {Kn : n ∈ N} be a collection of nonempty compact subsets of R N such that for all n, Kn+1 ⊂ Kn. Show that K = T∞ n=1 Kn is compact. Can K ever be the empty set?

Determine if there exist a nonempty set S with operation ⋆ on S and a nonempty...

Determine if there exist a nonempty set S with operation ⋆ on S and a nonempty set S′ ⊂ S, which is closed with respect to ⋆, satisfying the following properties. 1) S has identity e with respect to ⋆. ′ 2) e ∈/ S . 3) S′ has an identity with respect to ⋆.

Prove p → (q ∨ r), q → s, r → s ⊢ p → s

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that...

Let S ⊆ R and let G be an arbitrary isometry of R . Prove that the symmetry group of G(S) is isomorphic to the symmetry group of S. Hint: If F is a symmetry of S, what is the corresponding symmetry of G(S)?

Let A = {a1, a2, a3, . . . , an} be a nonempty set of...

Let A = {a1, a2, a3, . . . , an} be a nonempty set of n distinct natural numbers. Prove that there exists a nonempty subset of A for which the sum of its elements is divisible by n.

. Let φ : R → S be a ring homomorphism of R onto S. Prove...

. Let φ : R → S be a ring homomorphism of R onto S. Prove the following: J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.

Let S be a set and P be a property of the elements of the set,...

Let S be a set and P be a property of the elements of the set, such that each element either has property P or not. For example, maybe S is the set of your classmates, and P is "likes Japanese food." Then if s ∈ S is a classmate, he/she either likes Japanese food (so s has property P) or does not (so s does not have property P). Suppose Pr(s has property P) = p for a uniformly...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let...

Let (F, <) be an ordered field, let S be a nonempty subset of F, let c ∈ F, and for purposes of this problem let cS = {cx | x ∈ S}. (Do not use this notation outside this problem without defining what you mean by the notation.) Assume that c > 0. (i) Show that an element b ∈ F is an upper bound for S if and only if cb is an upper bound for cS. (ii)...

Subjects