# And why this theorem is important for us? Because many... are concerned with finding a maximal (or a minimal) value...

## Transcription

And why this theorem is important for us? Because many... are concerned with finding a maximal (or a minimal) value...
```8. SEQUENCES AND SUBSEQUENCES
11
And why this theorem is important for us? Because many economic problems
are concerned with finding a maximal (or a minimal) value of a function on some set.
Weierstrass theorem provides conditions under which such search is meaningful!!!
This theorem and its implications will be much dwelt upon later in the notes, so
we just give here one example. The consumer utility maximisation problem is the
problem of finding the maximum of utility function subject to the budget constraint.
According to Weierstrass theorem, this problem has a solution if utility function is
continuous and the budget set is compact.
8. Sequences and Subsequences
Let us consider again some metric space (X, d). An infinite sequence of points
in (X, d) is simply a list
x1 , x2 , x3 , . . . ,
where . . . indicates that the list continues “forever.”
numbers (or counting numbers) 1, 2, 3, . . . , which we denote N. We can now define
an infinite sequence in the following way.
Definition 6. An infinite sequence of elements of X is a function from N to
X.
Notation. If we look at the previous definition we see that we might have
a sequence s : N → X which would define s(1), s(2), s(3), . . . or in other words
would define s(n) for any natural number n. Typically when we are referring to
sequences we use subscripts (or sometimes superscripts) instead of parentheses and
write s1 , s2 , s3 , . . . and sn instead of s(1), s(2), s(3), . . . and s(n). Also rather than
saying that s : N → X is a sequence we say that {sn } is a sequence or even that
{sn }∞
n=1 is a sequence.
Lets now examine a few examples.
Example 4. Suppose that√(X, d) is R the real numbers with the usual metric
d(, x, y) = |x − y|. Then {n}, { n}, and {1/n} are sequences.
Example 5. Again, suppose that (X, d) is R the real numbers with the usual
metric d(x, y) = |x − y|. Consider the sequence {xn } where
(
1 if n is odd
xn =
0 if n is even
√
We see that {n} and { n} get arbitrary large as n gets larger, while in the last
example xn “bounces” back and forth between 0 and 1 as n gets larger. However for
{1/n} the element of the sequence gets closer and closer to 0 (and indeed arbitrarily
close to 0). We say, in this case, that the sequence converges to zero or that the
sequence has limit 0. This is a particularly important concept and so we shall give
a formal definition.
Definition 7. Let {xn } be a sequence of points in (X, d). We say that the
sequence converges to x0 ∈ X if for any ε > 0 there is N ∈ N such that if n > N
then d(xn , x0 ) < ε.
Informally we can describe this by saying that if n is large then the distance
from xn to x0 is small.
If the sequence {xn } converges to x0 , then we often write xn → x0 as n → ∞
or limn→∞ xn = x0 .
12
1. LOGIC, SETS, FUNCTIONS, AND SPACES
Exercise 34. Show that if the sequence {xn } converges to x0 then it does not
converge to any other value unequal to x0 . Another way of saying this is that if
the sequence converges then it’s limit is unique.
We have now seen a number of examples of sequences. In some the sequence
“runs off to infinity;” in others it “bounces around;” while in others it converges to
a limit. Could a sequence do anything else? Could a sequence, for example, settle
down each element getting closer and closer to all future elements in the sequence
but not converging to any particular limit? In fact, depending on what the space
X is this is indeed possible.
First let us recall the notion of a rational number. A rational number is a
number that can be expressed as the ratio of two integers, that is r is rational if
r = a/b with a and b integers and b 6= 0. We usually denote the set of all rational
numbers Q (since we have already used R for the real numbers). We now consider
and example in which the underlying space X is Q. Consider the sequence of
rational numbers defined in the following way
x1 = 1
xn + 2
.
xn+1 =
xn + 1
This kind of definition is called a recursive definition. Rather than writing, as a
function of n, what xn is we write what x1 is and then what xn+1 is as a function
of what xn is. We can obviously find any element of the sequence that we need, as
long as we sequentially calculate each previous element. In our case we’d have
x1 = 1
1+2
3
= = 1.5
x2 =
1+1
2
3
+
2
7
x3 = 23
= = 1.4
5
2 +1
x4 =
x5 =
x6 =
7
17
5 +2
=
≈ 1.416667
7
12
+
1
5
17
41
12 + 2
=
≈ 1.413793
17
29
12 + 1
41
99
29 + 2
≈ 1.414286
=
41
70
29 + 1
..
.
We see that the sequence goes up and down but that it seems to be “converging.” What is it converging to? Lets suppose that it’s converging to some value x0 .
Recall that
xn + 2
.
xn+1 =
xn + 1
We’ll see later that if f is a continuous function then lim n → ∞f (xn ) = f (lim n → ∞xn ).
In this case that means that
xn + 2
x0 = lim n → ∞xn+1 = lim n → ∞
xn + 1
x0 + 2
=
.
x0 + 1
Thus we have
x0 + 2
x0 =
x0 + 1
8. SEQUENCES AND SUBSEQUENCES
13
√
and if we solve this we obtain x0 = ±√ 2. Clearly if xn > 0 then
√ xn+1 >√0 so
our sequence can’t be converging to − 2 so we must have x0 = 2. But 2 is
not in Q. Thus we have a sequence of elements in Q that are getting very close to
each other but are not converging to any element of Q. (Of course the sequence is
converging to a point in R. In fact one construction of the real number system is
in terms of such sequences in Q.
Definition 8. Let {xn } be a sequence of points in (X, d). We say that the
sequence is a Cauchy sequence if for any ε > 0 there is N ∈ N such that if n, m > N
then d(xn , xm ) < ε.
Exercise 35. Show that if {xn } converges then {xn } is a Cauchy sequence.
A metric space (X, d) in which every Cauchy sequence converges to a limit in
X is called a complete metric space. The space of real numbers R is a complete
metric space, while the space of rationals Q is not.
Exercise 36. Is N the space of natural or counting numbers with metric d
given by d(x, y) = |x − y| a complete metric space?
In Section 6 we defined the notion of a function being continuous at a point.
It is possible to give that definition in terms of sequences.
Definition 9. Suppose (X, dX ) and (Y, dY ) are metric spaces, x0 ∈ X, and
f : X → Y is a function. Then f is continuous at x0 if for every sequence {xn } that
converges to x0 in (X, dX ) the sequence {f (xn )} converges to f (x0 ) in (Y, dY ).
Exercise 37. Show that the function f (x) = (x + 2)/(x + 1) is continuous at
any point x 6= −1. Show that this means that if xn → x0 as n → ∞ then
x0 + 2
xn + 2
=
.
lim
n→∞ xn + 1
x0 + 1
We can also define the concept of a closed set (and hence the concepts of open
sets and compact sets) in terms of sequences.
Definition 10. Let (X, d) be a metric space. A set S ⊂ X is closed if for any
convergent sequence {xn } such that xn ∈ S for all n then limn→∞ xn ∈ S. A set is
open if its complement is closed.
Given a sequence {xn } we can define a new sequence by taking only some of
the elements of the original sequence. In the example we considered earlier in which
xn was 1 if n was odd and 0 if n was even we could take only the odd n and thus
obtain a sequence that did converge. The new sequence is called a subsequence of
the old sequence.
Definition 11. Let {xn } be some sequence in (X, d). Let {nj }∞
j=1 be a
sequence of natural numbers such that for each j we have nj < nj+1 , that is
n1 < n2 < n3 < . . . . The sequence {xnj }∞
j=1 is called a subsequence of the original
sequence.
The notion of a subsequence is often useful. We often use it in the way that
we briefly referred to above. We initially have a sequence that may not converge,
but we are able to take a subsequence that does converge. Such a subsequence is
called a convergent subsequence.
Definition 12. A subset of a metric space with the property that every sequence in the subset has a convergent subsequence is called sequentially compact.
Theorem 3. In any metric space any compact set is sequentially compact.
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1. LOGIC, SETS, FUNCTIONS, AND SPACES
If we restrict attention to finite dimensional Euclidian spaces the situation is
even better behaved.
Theorem 4. Any subset of Rn is sequentially compact if and only if it is
compact.
Exercise 38. Verify the following limits.
n
(i) lim
=1
n→∞ n + 1
n+3
=0
(ii) lim 2
n→∞ √
n +1 √
(iii) lim n + 1 − n = 0
n→∞ √
n
(iv) lim
an + bn = max{a, b}
n→∞
Exercise 39. Consider a sequence {xn } in R. What can you say about the
sequence if it converges and for each n xn is an integer.
Exercise 40. Consider the sequence
1 1 2 1 2 3 1 2 3 4 1
2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, . . . .
For which values z ∈ R is there a subsequence converging to z?
Exercise 41. Prove that if a subsequence of a Cauchy sequence converges to
a limit z then so does the original Cauchy sequence.
Exercise 42. Prove that any subsequence of a convergent sequence converges.
Finally one somewhat less trivial exercise.
Exercise 43. Prove that if limn→∞ xn = z then
x1 + · · · + xn
=z
lim
n→∞
n
9. Linear Spaces
The notion of linear space is the axiomatic way of looking at the familiar linear
operations: addition and multiplication. A trivial example of a linear space is the
set of real numbers, R.
What is the operation of addition? The one way of answering the question is
saying that the operation of addition is just the list of its properties. So, we will
define the addition of elements from some set X as the operation that satisfies the
following four axioms.
A1: x + y = y + x for all x and y in X.
A2: x + (y + z) = (x + y) + z, for all x, y, and z in X.
A3: There exists an element, denoted by 0, such that x + 0 = x for all x in
X.
A4: For every x in X there exist an element y in X, called inverse of x, such
that x + y = 0.
And, to make things more interesting we will also introduce the operation of
‘multiplication by number’ by adding two more axioms.
A5: 1x = x for all x in X.
A6: α(βx) = (αβ)x for all x in X and for all α and β in R.
Finally, two more axioms relating addition and multiplication.
A7: α(x + y) = αx + αy for all x and y in X and for all α in R.
A8: (α + β)x = αx + βx for all x in X and for all α and β in R.
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