Chapter 2 - Sequences Of Numbers Flashcards
Is the number a sequence converges to unique ?
Yes
What’s the difference between an increasing sequence and a strictly increasing sequence?
An increasing sequence requires xn be less than or equal to xn+1 for all positive integers of n. A strictly increasing sequence requires xn be STRICTLY less than xn+1
State theorem 1.1 … Dealing with an increasing set bounded from above.
Let {xn} be an increasing sequence, bounded from above. Then the least upperbound - b - of the sequence {xn} is the limit of the sequence.
A decreasing sequence is bounded from below, what is the limit of the sequence?
The greatest lower bound is the limit of the sequence.
When is a sequence Cauchy?
If, given some positive epsilon, there exists N such that for all m,n greater than or equal to N we have |xm-xn|
If a sequence converges is it Cauchy?
Yes.
If a sequence is Cauchy does it converge?
Yes.
Are all Cauchy sequences bounded?
Yes.
What is a point of accumulation?
Let {xn} be a sequence. If given epsilon there exists infinitely many integers such that |xn-x| is less than epsilon then x is a point of accumulation.
State the W-B theorem.
Let {xn} be a sequence, and let a, b be numbers such that a
What does every bounded sequence of real numbers have? (Theorem 1.5)
A convergent subsequence. If the bounded set lies between [a,b] the so does the limit of the subsequence.
What is a function?
A function is a map from some set S, into the real numbers.
What do we mean by adherent?
Let S be a set of numbers. Let a be a number. a is adherent TO S if given epsilon, there exists an element x in S such that |x-a|
If a set consists of a single number how many adherent points does it have?
1, that single number.
Is the least upper bound of a non empty set S adherent?
Yes .
