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Questions and Answers

What characterizes a sequence as strictly increasing?

  • an+1 > an for all n ∈ N (correct)
  • an+1 < an for all n ∈ N
  • an+1 = an for any n ∈ N
  • an is constant for all n ∈ N

Which of the following sequences is monotonic?

  • an = sin(n)
  • an = n + 2 (correct)
  • an = n^2 (correct)
  • an = (-1)^n

What is the definition of a sequence that is bounded above?

  • an approaches infinity as n approaches infinity
  • There exists M ∈ R such that an ≤ M for all n ∈ N (correct)
  • an < 0 for all n ∈ N
  • There exists M ∈ R such that an ≥ M for all n ∈ N

Which condition indicates that a sequence is not bounded below?

<p>The sequence has terms that can go below any selected M (B)</p> Signup and view all the answers

What is an example of a strictly decreasing sequence?

<p>an = -n (D)</p> Signup and view all the answers

In the context of the given definitions, which statement is true about the sequence an = -n?

<p>All of the above (D)</p> Signup and view all the answers

What is meant by a monotonic sequence?

<p>A sequence that is either strictly increasing or strictly decreasing (C)</p> Signup and view all the answers

Which of the following characteristics does NOT define a bounded sequence?

<p>The sequence oscillates infinitely (D)</p> Signup and view all the answers

What is the condition for a sequence (an) to be classified as bounded?

<p>There exists M ∈ R such that for all n ∈ N, |an| ≤ M. (C)</p> Signup and view all the answers

Which of the following sequences is an example of a bounded sequence?

<p>an = 1/n (A), an = sin(n) (B)</p> Signup and view all the answers

What does it mean for a sequence (an) to be unbounded?

<p>For every real number M, there exists an n such that |an| &gt; M. (D)</p> Signup and view all the answers

If a sequence is both bounded above and bounded below, what can we deduce about the sequence?

<p>The sequence is bounded. (B)</p> Signup and view all the answers

In terms of logic, how can the statement 'a sequence (an) is unbounded' be formulated?

<p>It is not true that (an) is bounded above and bounded below. (C)</p> Signup and view all the answers

Why can we not determine boundedness for a sequence of complex numbers using inequalities?

<p>Complex numbers cannot be ordered in the same way as real numbers. (A)</p> Signup and view all the answers

Which of the following best describes the sequence an = -n?

<p>It is unbounded. (B)</p> Signup and view all the answers

What is necessary for a sequence (an) of complex numbers to be considered bounded?

<p>The absolute values |an| must all be less than some constant M. (D)</p> Signup and view all the answers

What must be true for a subset of complex numbers A to be considered bounded?

<p>There exists an M ∈ R such that for all z ∈ A, |z| ≤ M. (C)</p> Signup and view all the answers

If a set A of complex numbers is bounded above, what can be inferred about its elements?

<p>There exists a real number M1 such that all z ∈ A satisfy z ≤ M1. (C)</p> Signup and view all the answers

Which inequality correctly describes the boundedness of a set A that is both bounded above and below?

<p>-M ≤ x ≤ M for all x ∈ A. (B)</p> Signup and view all the answers

What contradiction arises when proving that A = {z ∈ C | Re(z) ≤ 1} is unbounded?

<p>We assumed there exists M such that |z| ≤ M but found a z where |z| &gt; M. (C)</p> Signup and view all the answers

What does the Triangle Inequality imply when considering the boundedness of the set A = {z ∈ C | |z - z0| < 1}?

<p>|z| &lt; 1 + |z0| for all z ∈ A. (B)</p> Signup and view all the answers

What is the primary characteristic of a sequence?

<p>It is an infinite list of numbers that can include rational and irrational values. (B)</p> Signup and view all the answers

Which property must a convergent sequence possess?

<p>The terms of the sequence must be bounded above. (B)</p> Signup and view all the answers

What does the proof show about the relationship between boundedness in sequences and the completeness property of the real numbers?

<p>Every bounded sequence has a limit in the real numbers. (D)</p> Signup and view all the answers

Flashcards

Bounded Set (Complex Numbers)

A set of complex numbers where all its elements have a modulus (distance from the origin) less than or equal to some real number M.

Upper Bound

A real number M such that every element in a set is less than or equal to M.

Lower Bound

A real number M such that every element in a set is greater than or equal to M.

Triangle Inequality (Complex)

For any two complex numbers z1 and z2, the distance |z1 + z2| is less than or equal to |z1| + |z2|.

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Unbounded Set

A set of complex numbers that is not bounded, meaning there's no single maximum distance from the origin.

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Modulus (Complex Number)

The distance of a complex number from the origin in the complex plane.

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Sequence

An ordered list of numbers, possibly infinite.

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Real Number

A number that can be represented on a number line (including rational and irrational numbers).

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Bounded Sequence

A sequence of real numbers where all terms lie within a certain distance from zero. There exists a real number M such that the absolute value of every term in the sequence is less than or equal to M.

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Strictly Increasing Sequence

A sequence where each term is greater than the previous term. This means an+1 > an for all natural numbers n.

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Strictly Decreasing Sequence

A sequence where each term is less than the previous term. This means an+1 < an for all natural numbers n.

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Unbounded Sequence

A sequence of real numbers that is NOT bounded. For any real number M, there's at least one term in the sequence whose absolute value is greater than M.

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Monotonic Sequence

A sequence that is either strictly increasing or strictly decreasing. It can't change its direction.

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Bounded Above

A sequence for which there exists a real number M such that every term in the sequence is less than or equal to M.

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Bounded Above Sequence

A sequence where all terms are less than or equal to a specific value (M). This means there exists a real number M such that an ≤ M for all natural numbers n.

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Bounded Below

A sequence for which there exists a real number M such that every term in the sequence is greater than or equal to M.

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Bounded Below Sequence

A sequence where all terms are greater than or equal to a specific value (M). This means there exists a real number M such that an ≥ M for all natural numbers n.

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Complex Number Modulus

The distance of a complex number from the origin in the complex plane.

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Bounded Sequence (Complex)

A sequence of complex numbers where there exists a real number M such that the modulus of every term in the sequence is less than or equal to M.

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Bounded Sequence

A sequence that is both bounded above and bounded below. Therefore, there exist real numbers M1 and M2 such that M1 ≤ an ≤ M2 for all natural numbers n.

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Sequence Boundedness

A sequence is bounded if and only if it is both bounded above and bounded below.

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What does 'an' represent in a sequence (an)?

'an' represents the general term of the sequence. It represents any specific element within the sequence, depending on the value of 'n'.

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What does 'n' represent in a sequence (an)?

'n' represents the position of a term in the sequence. It's a natural number that indicates which term we are looking at in the sequence.

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Equivalence of Boundedness Statements

The statement that a sequence is bounded is equivalent to the statement that it is both bounded above and bounded below.

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Study Notes

Mathematical Logic and Sets

  • Propositions: A statement that is either True or False.

  • Examples of Propositions:

    • 1 + 1 = 2
    • π = 3
    • There exists a real number x such that x2 = x
    • All even integers greater than 2 can be written as the sum of two primes.

  • Predicates: A statement that depends on one or more variables and becomes a proposition when values are assigned to these variables.

  • Examples of Predicates:

    • n is a prime number
    • n+m < 4
    • x2 > y2
    • x2 + 3x + 2 = 0

Logical Connectives

  • OR: A statement 'P or Q' is True precisely when one or both of P and Q are True.

  • AND: A statement 'P and Q' is True precisely when both P and Q are True.

  • Truth Tables: Used to summarize the truth values of various logical combinations of propositions.

Implications

  • P implies Q: True in all cases except when P is True and Q is False.
  • Denotation: P)Q

Sets

  • Definition: A collection of objects.
  • Elements: Objects within a set.
  • Notation: x ∈ A means 'x is an element of A' and x /2 A means 'x is not an element of A'.
  • Finite Sets: Sets with a limited number of elements.
  • Infinite Sets: Sets with an unlimited number of elements.
  • Empty Set: A set with no elements. Denoted by ∅
  • Standard Number Sets:
    • Natural Numbers (N): {1, 2, 3, ...}
    • Integers (Z): {..., -2, -1, 0, 1, 2, ...}
    • Rational Numbers (Q): {a/b | a, b ∈ Z, b ≠ 0}
    • Real Numbers (R): Includes all rational and irrational numbers.
    • Complex Numbers (C): {a + bi | a, b ∈ R}

Subsets

  • Definition: A set A is a subset of a set B (A ⊆ B) if every element of A is also an element of B.
  • Proper Subset A is a proper subset of B (A ⊂ B) if A is a subset of B and A is not equal to B.

Operations on Sets

  • Intersection (A ∩ B): The set of elements that are in both A and B.
  • Union (A ∪ B): The set of elements that are in A or B (or both).
  • Difference (A \ B): The set of elements that are in A but not in B.
  • Complement (Ac): The set of elements that are in the universal set U but not in A.

Quantifiers

  • There exists (∃): Means 'there is at least one'.
  • For all (∀): Means 'for every'.

Methods of Proof

  • Direct Proof: Starts with assumptions and uses implications to reach a conclusion.
  • Proof by Contradiction: Assumes a statement is false and derives a contradiction to prove the statement is true.

Proof by Induction

  • Base Case: Proving the statement is true for the first case (often n=1).
  • Inductive Step: Assuming the statement is true for some value k and proving that it is also true for k+1. This proves that the statement holds for all natural numbers n.

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