Question

True or False: Any finite set of real numbers is complete. Either prove or provide a counterexample.

Answer #1

False statement.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of nonnegative real numbers.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of non-negative real numbers.

1) Prove: Any real number is an accumulation point of the set of
rational number.
2) prove: if A ⊆ B and A,B are bounded then supA ≤ supB .
3) Give counterexample: For two sequences {an} and
{bn}, if {anbn} converges then
both sequences are convergent.

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

3. For each of the following statements, either provide a short
proof that it is true (or appeal to the deﬁnition) or provide a
counterexample showing that it is false.
(e) Any set containing the zero vector is linearly
dependent.
(f) Subsets of linearly dependent sets are linearly
dependent.
(g) Subsets of linearly independent sets are linearly
independent.
(h) The rank of a matrix is equal to the number of its nonzero
columns.

(a) Is the converse of Bolzano-Weierstrass Theorem true? If yes
prove it. If false provide a counterexample.
(b) Since Q is countably infinite, it can be written as a sequence
{xn}. Can {xn} be monotone? Briefly
explain. Hint. Assume it’s monotone, what would be the
consequences?
(c) Use the , N definition to prove that if {xn} and {yn} are
Cauchy then {xn + yn} is Cauchy too.

Prove the conjecture or provide a counterexample:
Let U ∈ in the usual topology, and let F be a finite set. Then
(U−F) ∈ in the usual topology.

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

Find a bijection between set of infinite subsets of natural
numbers and real numbers.
Find a bijection between set of finite subsets of real numbers
and real numbers.
Find a bijection between set of countable subsets of real
numbers and real numbers.

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

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