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.
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 ⋆.
You are given a set S of n real numbers where for each element
x ∈ S, 1 < | x | < 10. You are also given a black box that
returns true if there is some combination (x, y, z) ∈ S such that x
· y = z.
Say the black box returns only z. Write an O(n log n)
algorithm to find x and y. In other words, given a set S with n real
numbers...
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?
PROOFS:
1. State the prove The Density Theorem for Rational Numbers
2. Prove that irrational numbers are dense in the set of real numbers
3. Prove that rational numbers are countable
4. Prove that real numbers are uncountable
5. Prove that square root of 2 is irrational
Real Analysis: Prove a subset of the Reals is compact if and only
if it is closed and bounded. In other words, the set of reals
satisfies the Heine-Borel property.
Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...
1. Prove that the Cantor set contains no intervals.
2. Prove: If x is an element of the Cantor set, then there is a
sequence Xn of elements from the Cantor set converging
to x.