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 2 years ago

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?

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

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 ⋆.

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 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 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 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...

Using the definition of Compact Set, prove that the union of two compact sets is compact....

Using the definition of Compact Set, prove that the union of two compact sets is compact. Use this result to show that the union of a finite collection of compact sets is compact. Is the union of any collection of compact sets compact?