Sequences Definitions

Front 1

Let S be any non-empty subset of the reals. What does it mean for a set S to be bounded above?

Back 1

S is bounded above is there is a real number M such that for all x in S, x is less than or equal to M.

Front 2

Let S be any non-empty subset of the reals. What does it mean for a set S to be bounded below?

Back 2

S is bounded below if there is a real number m such that for all x in S, x is greater than or equal to m.

Front 3

Let S be any non-empty subset of the reals and assume S is bounded above. What is a supremum?

Back 3

A supremum is a real number, say M, such that M is an upper bound for S and if M' is a real number for which M' < M, then M' is not an upper bound for S.

Front 4

State the completeness property.

Back 4

Any non-empty subset of the reals which is bounded above has a supremum.

Front 5

Assume S is a non-empty subset of the reals and that S is bounded below. What is an infimum?

Back 5

An infimum of S is a real number, say m, such that m is a lower bound for S and if m' is a real number and m

Front 6

What is a sequence?

Back 6

A sequence is a list a1, a2, ..., an, ... of real numbers labelled by natural numbers. an is the nth term of the sequence (an)n{naturals}.

Front 7

What is |x|?

Back 7

|x|=x if x is greater than or equal to 0

|x|=-x is x is less than 0

Front 8

What does it mean for a sequence to converge to a real number l?

Back 8

A sequence converges to the real number l if for every E>0, there exists an N in the naturals such that for every n greater than or equal to N, |an-l|

Front 9

What does it mean for a sequence to be bounded?

Back 9

A sequence is called bounded if there exists a positive real value, say M, such that for every n in the naturals, |an| is less than or equal to M.

Front 10

What does it mean for a sequence to be monotonic?

Back 10

A sequence is said to be monotonic if it is either increasing or decreasing (either generally or strictly).

Front 11

What does it mean for a sequence to be increasing?

Back 11

A sequence is said to be increasing if for all n in the naturals, an is less than or equal to a(n+1).

Front 12

What does it mean for a sequence to be decreasing?

Back 12

A sequence is said to be decreasing if for all n in the naturals, a(n+1) is less than or equal to an.

Front 13

What does it mean for a sequence to be strictly increasing?

Back 13

A sequence is said to be strictly increasing if for all n in the naturals, an is less than a(n+1).

Front 14

What does it mean for a sequence to be strictly decreasing?

Back 14

A sequence is said to be strictly decreasing if for all n in the naturals, a(n+1) is less than an.

Front 15

What is a null sequence?

Back 15

A null sequence is a sequence such that an tends to 0 as n tends to infinity.

Front 16

What does it mean for a sequence to be divergent?

Back 16

A sequence is divergent if it is not convergent, i.e. there is no real value l such than an tends to l as n tends to infinity.

Front 17

What does it mean for a sequence to tend to ininfinity?

Back 17

A sequence tends to infinity if for each positive real number K, there exists an integer N such that for all n greater than or equal to K, an>K. Then an tends to infinity as n tends to infinity.

Front 18

What does it mean for a sequence to tend to minus infinity?

Back 18

A sequence tends to minus infinity if for each negative real number K, there exists an integer N such that for all n greater than or equal to N, an

Front 19

What is a subsequence of a sequence (an)?

Back 19

A subsequence of (an) is a sequence such that if 1 is less than or equal to k1 which is less than or equal to k2 which is strictly less than k2 ... strictly less than kn ... is a strictly increasing sequence of naturals numbers, then given any sequence (an) of real numbers we can form the sequence (akn) - a sequence formed by taking out terms of the original sequence (so long as infinitely many terms are left).

Front 20

What is a Cauchy sequence?

Back 20

A sequence (an) is called a Cauchy sequence if for all E> 0, there exists N ∈N such that for all i,j in the naturals with i,j greater than or equal to N, we have |ai −aj|<E.

So the terms are getting closer and closer together.