Question

Complex Analysis Proof - Prove: A set is closed if and only if S contains all of its accumulation points.

Answer #1

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...

Complex Analysis Proof - Prove the uniqueness of the limit.

Complex Analysis Proof - Prove: if f = u + iv is analytic in a
domain D, then u and v satisfy the Cauchy-Riemann equations in
D.

Prove the statements (a) and (b) using a set element proof and
using only the definitions of the set operations (set equality,
subset, intersection, union, complement):
(a) Suppose that A ⊆ B. Then for every set C, C\B ⊆ C\A.
(b) For all sets A and B, it holds that A′ ∩(A∪B) = A′ ∩B.
(c) Now prove the statement from part (b)

Prove that a closed set in the Zariski topology on K1 is either
the empty set, a finite collection of points, or K1 itself.

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Proof: A graph is Eulerian if and only if it has an Eulerian
tour. (i.e. Closed Eulerian trail which visits every edge exactly
once.)
Please show full detailed proof. Thank you in advance!

Prove or provide a counterexample
If A is a nonempty countable set, then A is closed in T_H.

Let S denote the set of all possible finite binary strings, i.e.
strings of finite length made up of only 0s and 1s, and no other
characters. E.g., 010100100001 is a finite binary string but
100ff101 is not because it contains characters other than 0, 1.
a. Give an informal proof arguing why this set should be
countable. Even though the language of your proof can be informal,
it must clearly explain the reasons why you think the set should...

prove that a compact set is closed using the Heine - Borel
theorem

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