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

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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.

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.

The goal is to show that a nonempty subset C⊆R is closed iff there is a...

The goal is to show that a nonempty subset C⊆R is closed iff there is a continuous function g:R→R such that C=g−1(0). 1) Show the IF part. (Hint: explain why the inverse image of a closed set is closed.) 2) Show the ONLY IF part. (Hint: you may cite parts of Exercise 4.3.12 if needed.)

let G be a simple graph. show that the relation R on the set of vertices...

let G be a simple graph. show that the relation R on the set of vertices of G such that URV if and only if there is an edge associated with (u,v) is a symmetric irreflexive relation on G

Let f : [a, b] → R be a monotone function. Show that the discontinuity set...

Let f : [a, b] → R be a monotone function. Show that the discontinuity set Disc(f) is countable.

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 ⋆.

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

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