Sequence
A sequence is one of the earliest and easiest definitions encountered, but I will restate it.
I was taught to denote the sequence {a1,a2,...} by {an}∞n=1 however I don't like this, as it looks like a set. I have seen the notation (an)∞n=1 and I must say I prefer it. This notation is inline with that of a tuple which is a generalisation of an ordered pair.
Contents
[hide]Definition
Formally a sequence (Ai)∞i=1 is a function[1][2], f:N→S where S is some set. For a finite sequence it is simply f:{1,...,n}→S. Now we can write:
- f(i):=Ai
This naturally then generalises to indexing sets
Notation
To specify that the points of a sequence, the xi are from a space, X we may write:
- (xn)∞n=1⊆X
This is an abuse of notation, as (xn)∞n=1 is not a subset of X. It plays on:
- [(xn)∞n=1⊆X]⟺[x∈(xn)∞n=1⟹x∈X]
Note that the elements of (xn)∞n=1 are ether:
- Elements of a relation (if we consider the sequence as a mapping) or
- So using this, x∈(xn)∞n=1 may look like x=(a,b) (indicating f(a)=b) which is an Ordered pair, not in X
- Elements of a tuple (which is a generalisation of ordered pairs where (usually) (a,b)={{a},{a,b}}
- So using this, x∈(xn)∞n=1 may indeed look like x={{a},{a,b}}∉X
As such the notation (xn)∞n=1⊆X having no other sensible meaning is a notation to say that ∀i[xi∈X]
Subsequence
Given a sequence (xn)∞n=1 we define a subsequence of (xn)∞n=1[3][4] as follows:
- Given any strictly increasing monotonic sequence[Note 1], (kn)∞n=1⊆N
- That means that ∀n∈N[kn<kn+1][Note 2]
Then the subsequence of (xn) given by (kn) is:
- (xkn)∞n=1, the sequence whose terms are: xk1,xk2,…,xkn,…
- That is to say the ith element of (xkn) is the kith element of (xn)
As a mapping
Consider an (injective) mapping: k:N→N with the property that:
- ∀a,b∈N[a<b⟹k(a)<k(b)]
This defines a sequence, (kn)∞n=1 given by kn:=k(n)
- Now (xkn)∞n=1 is a subsequence
See also
- Subsequence
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy sequence (Alternatively: Cauchy criterion for convergence)
- Convergence of a sequence (Or Limit (sequence) - the page Convergence of a sequence is being refactored into it)
Notes
- Jump up ↑ Note that strictly increasing cannot be replaced by non-decreasing as the sequence could stay the same (ie a term where mi=mi+1 for example), it didn't decrease, but it didn't increase either. It must be STRICTLY increasing.
If it was simply "non-decreasing" or just "increasing" then we could define: kn:=5 for all n.- Then (xkn)n∈N is a constant sequence where every term is x5 - the 5th term of (xn).
- Jump up ↑ Some books may simply require increasing, this is wrong. Take the theorem from Equivalent statements to compactness of a metric space which states that a metric space is compact ⟺ every sequence contains a convergent subequence. If we only require that:
- kn≤kn+1
The mapping definition directly supports this, as the mapping can be thought of as choosing terms
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- Jump up ↑ p11 - Analysis - Part 1: Elements - Krzysztof Maurin
- Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
- Jump up ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha